# Distinct Selection Eligibility and Fixed Prefixes

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261011-advance-20261015-r06-permutation-combination-foundations · 2026-10-11 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=0, k=0, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(1, 1, 1, Yes)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (1, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0001/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 2. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=1, k=1, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(1, 1, 1, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (1, 1, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0002/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 3. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20, k=19, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(2432902008176640000, 20, 121645100408832000, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (2432902008176640000, 20, 121645100408832000, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0003/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 4. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=4, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(24, 4, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (24, 4, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0004/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 5. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=5, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(20, 10, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (20, 10, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0005/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 6. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=6, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(120, 20, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (120, 20, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0006/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 7. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=7, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(42, 21, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (42, 21, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0007/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 8. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=8, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(336, 56, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (336, 56, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0008/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 9. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=9, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(72, 36, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (72, 36, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0009/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 10. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=10, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(720, 120, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (720, 120, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0010/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 11. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=11, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(110, 55, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (110, 55, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0011/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 12. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=12, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(1320, 220, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (1320, 220, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0012/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 13. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=13, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(156, 78, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (156, 78, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0013/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 14. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=14, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(2184, 364, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (2184, 364, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0014/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 15. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=15, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(210, 105, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (210, 105, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0015/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 16. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=16, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(3360, 560, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (3360, 560, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0016/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 17. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=17, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(272, 136, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (272, 136, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0017/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 18. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=18, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(4896, 816, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (4896, 816, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0018/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 19. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=19, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(342, 171, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (342, 171, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0019/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 20. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(6840, 1140, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (6840, 1140, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0020/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 21. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=21, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(420, 210, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (420, 210, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0021/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 22. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=22, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(9240, 1540, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (9240, 1540, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0022/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 23. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=23, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(506, 253, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (506, 253, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0023/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 24. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=24, k=3, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(12144, 2024, 6, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (12144, 2024, 6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0024/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 25. Ordered/Unordered Counts, Factorial, and Empty Selection

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=25, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

View answer and explanation(600, 300, 2, No)

Compute the unordered count exactly and multiply by the ordering factor for the ordered count; an empty selection and empty ordering each count once. The exact result, retaining the requested order, is (600, 300, 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0025/)

Permutations and combinations · qf-density15-selection-order-factor-triple · candidate





## 26. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(0)=∅, k=0, candidates=[[], [0], [0, 0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, No, No], [Yes, No, No], [Yes, Yes, No], [Yes, No, No], [[], invalid-selection, invalid-selection], [[], invalid-selection, invalid-selection])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, No, No], [Yes, No, No], [Yes, Yes, No], [Yes, No, No], [[], invalid-selection, invalid-selection], [[], invalid-selection, invalid-selection]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0026/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 27. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(4)={0,…,3}, k=2, candidates=[[1, 0], [0, 1], [0, 0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, No], [Yes, Yes, No], [[1, 0], [0, 1], invalid-selection], [[0, 1], [0, 1], invalid-selection])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, No], [Yes, Yes, No], [[1, 0], [0, 1], invalid-selection], [[0, 1], [0, 1], invalid-selection]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0027/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 28. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(4)={0,…,3}, k=2, candidates=[[0], [0, 1, 2], [0, 4]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([No, No, Yes], [Yes, Yes, No], [Yes, Yes, Yes], [No, No, No], [invalid-selection, invalid-selection, invalid-selection], [invalid-selection, invalid-selection, invalid-selection])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, No, Yes], [Yes, Yes, No], [Yes, Yes, Yes], [No, No, No], [invalid-selection, invalid-selection, invalid-selection], [invalid-selection, invalid-selection, invalid-selection]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0028/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 29. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0, 1], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [[0, 1], [0, 1], [0, 1]], [[0, 1], [0, 1], [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [[0, 1], [0, 1], [0, 1]], [[0, 1], [0, 1], [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0029/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 30. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0, 1], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([No, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0030/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 31. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 0], [0, 1], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0031/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 32. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0, 1], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, Yes], [No, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, Yes, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [invalid-selection, [0, 1], [0, 1]], [invalid-selection, [0, 1], [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0032/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 33. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, No, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, No, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0033/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 34. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([No, No, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, No, Yes], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0034/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 35. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 0], [0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, No, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, No, Yes], [Yes, Yes, Yes], [No, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0035/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 36. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, No, Yes], [No, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, No, Yes], [No, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0036/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 37. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0, 0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0037/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 38. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0, 0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([No, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, Yes, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0038/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 39. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 0], [0, 0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0039/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 40. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0, 0], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, Yes], [No, Yes, Yes], [Yes, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, Yes, Yes], [Yes, No, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0040/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 41. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0, 8], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, Yes], [Yes, No, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, No, Yes], [Yes, Yes, Yes], [Yes, No, Yes], [[0, 1], invalid-selection, [0, 1]], [[0, 1], invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0041/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 42. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0, 8], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([No, Yes, Yes], [Yes, No, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, Yes, Yes], [Yes, No, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0042/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 43. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 0], [0, 8], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0043/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 44. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0, 8], [0, 1]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, Yes], [No, No, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, No, Yes], [Yes, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0044/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 45. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0, 1], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, No], [[0, 1], [0, 1], invalid-selection], [[0, 1], [0, 1], invalid-selection])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, Yes, No], [[0, 1], [0, 1], invalid-selection], [[0, 1], [0, 1], invalid-selection]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0045/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 46. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0, 1], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([No, Yes, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, Yes, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0046/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 47. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 0], [0, 1], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, No], [Yes, Yes, Yes], [No, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, No], [Yes, Yes, Yes], [No, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0047/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 48. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0, 1], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, Yes, No], [No, Yes, Yes], [Yes, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, Yes, No], [No, Yes, Yes], [Yes, Yes, Yes], [No, Yes, No], [invalid-selection, [0, 1], invalid-selection], [invalid-selection, [0, 1], invalid-selection]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0048/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 49. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 1], [0], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([Yes, No, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, No], [[0, 1], invalid-selection, invalid-selection], [[0, 1], invalid-selection, invalid-selection])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([Yes, No, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, No], [[0, 1], invalid-selection, invalid-selection], [[0, 1], invalid-selection, invalid-selection]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0049/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 50. Candidate Selection Length, Typing, and Distinctness

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0], [0], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.

View answer and explanation([No, No, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, No], [invalid-selection, invalid-selection, invalid-selection], [invalid-selection, invalid-selection, invalid-selection])

Check length, typing, and distinctness as separate axes; only an eligible row can produce the ordered tuple and its unordered version. The exact result, retaining the requested order, is ([No, No, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, No], [invalid-selection, invalid-selection, invalid-selection], [invalid-selection, invalid-selection, invalid-selection]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0050/)

Permutations and combinations · qf-density15-selection-fixed-label-qualification · candidate





## 51. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=1≥1, k=0, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(1, 0, 1, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (1, 0, 1, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0051/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 52. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=3≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(1, 1, 0, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (1, 1, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0052/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 53. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20≥1, k=10, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(184756, 92378, 92378, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (184756, 92378, 92378, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0053/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 54. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=6≥1, k=5, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(6, 5, 1, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (6, 5, 1, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0054/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 55. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=5≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(10, 4, 6, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (10, 4, 6, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0055/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 56. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=6≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(20, 10, 10, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (20, 10, 10, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0056/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 57. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=7≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(21, 6, 15, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (21, 6, 15, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0057/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 58. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=8≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(56, 21, 35, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (56, 21, 35, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0058/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 59. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=9≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(36, 8, 28, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (36, 8, 28, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0059/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 60. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=10≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(120, 36, 84, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (120, 36, 84, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0060/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 61. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=11≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(55, 10, 45, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (55, 10, 45, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0061/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 62. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=12≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(220, 55, 165, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (220, 55, 165, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0062/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 63. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=13≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(78, 12, 66, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (78, 12, 66, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0063/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 64. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=14≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(364, 78, 286, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (364, 78, 286, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0064/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 65. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=15≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(105, 14, 91, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (105, 14, 91, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0065/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 66. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=16≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(560, 105, 455, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (560, 105, 455, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0066/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 67. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=17≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(136, 16, 120, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (136, 16, 120, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0067/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 68. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=18≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(816, 136, 680, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (816, 136, 680, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0068/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 69. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=19≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(171, 18, 153, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (171, 18, 153, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0069/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 70. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(1140, 171, 969, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (1140, 171, 969, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0070/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 71. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=21≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(210, 20, 190, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (210, 20, 190, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0071/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 72. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=22≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(1540, 210, 1330, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (1540, 210, 1330, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0072/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 73. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=23≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(253, 22, 231, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (253, 22, 231, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0073/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 74. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=24≥1, k=3, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(2024, 253, 1771, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (2024, 253, 1771, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0074/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 75. Combinations Including or Excluding a Fixed Existing Label

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=25≥1, k=2, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole).

View answer and explanation(300, 24, 276, 0)

Remove the fixed object and count the two eligible subset sizes separately; k=0 has no selection containing 0. The exact result, retaining the requested order, is (300, 24, 276, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0075/)

Permutations and combinations · qf-density15-selection-one-distinguished-counts · candidate





## 76. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=0, k=0, r=0; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(0, 0, 1, 1, 1/1)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (0, 0, 1, 1, 1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0076/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 77. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=4, k=2, r=0; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(4, 2, 12, 12, 1/1)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (4, 2, 12, 12, 1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0077/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 78. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=4, k=2, r=2; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(2, 0, 1, 12, 1/12)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (2, 0, 1, 12, 1/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0078/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 79. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=32, k=32, r=16; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(16, 16, 20922789888000, 263130836933693530167218012160000000, 1/12576278705767096320000)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (16, 16, 20922789888000, 263130836933693530167218012160000000, 1/12576278705767096320000).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0079/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 80. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=5, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(4, 2, 12, 60, 1/5)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (4, 2, 12, 60, 1/5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0080/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 81. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=6, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(5, 2, 20, 120, 1/6)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (5, 2, 20, 120, 1/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0081/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 82. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=7, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(6, 2, 30, 210, 1/7)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (6, 2, 30, 210, 1/7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0082/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 83. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=8, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(7, 2, 42, 336, 1/8)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (7, 2, 42, 336, 1/8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0083/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 84. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=9, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(8, 2, 56, 504, 1/9)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (8, 2, 56, 504, 1/9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0084/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 85. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=10, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(9, 2, 72, 720, 1/10)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (9, 2, 72, 720, 1/10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0085/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 86. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=11, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(10, 2, 90, 990, 1/11)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (10, 2, 90, 990, 1/11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0086/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 87. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=12, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(11, 2, 110, 1320, 1/12)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (11, 2, 110, 1320, 1/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0087/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 88. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=13, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(12, 2, 132, 1716, 1/13)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (12, 2, 132, 1716, 1/13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0088/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 89. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=14, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(13, 2, 156, 2184, 1/14)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (13, 2, 156, 2184, 1/14).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0089/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 90. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=15, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(14, 2, 182, 2730, 1/15)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (14, 2, 182, 2730, 1/15).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0090/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 91. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=16, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(15, 2, 210, 3360, 1/16)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (15, 2, 210, 3360, 1/16).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0091/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 92. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=17, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(16, 2, 240, 4080, 1/17)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (16, 2, 240, 4080, 1/17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0092/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 93. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=18, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(17, 2, 272, 4896, 1/18)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (17, 2, 272, 4896, 1/18).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0093/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 94. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=19, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(18, 2, 306, 5814, 1/19)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (18, 2, 306, 5814, 1/19).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0094/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 95. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=20, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(19, 2, 342, 6840, 1/20)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (19, 2, 342, 6840, 1/20).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0095/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 96. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=21, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(20, 2, 380, 7980, 1/21)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (20, 2, 380, 7980, 1/21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0096/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 97. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=22, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(21, 2, 420, 9240, 1/22)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (21, 2, 420, 9240, 1/22).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0097/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 98. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=23, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(22, 2, 462, 10626, 1/23)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (22, 2, 462, 10626, 1/23).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0098/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 99. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=24, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(23, 2, 506, 12144, 1/24)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (23, 2, 506, 12144, 1/24).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0099/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate





## 100. Completions after Fixing an Initial Ordered Prefix

Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=25, k=3, r=1; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio).

View answer and explanation(24, 2, 552, 13800, 1/25)

Remove the used objects and filled positions before computing both ordered counts and reducing their ratio; a counting ratio does not automatically assert a probability model. The exact result, retaining the requested order, is (24, 2, 552, 13800, 1/25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r06-permutation-combination-0100/)

Permutations and combinations · qf-density15-selection-forced-initial-order · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261011-advance-20261015-r06-permutation-combination-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261011-advance-20261015-r06-permutation-combination-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261011-advance-20261015-r06-permutation-combination-foundations/validation-report.json)