Distinct Selection Eligibility and Fixed Prefixes
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Permutations and combinations · qf-density15-selection-forced-initial-order · candidate
Canonical question JSONL · English JSONL · Validation report