# Permutation and Combination Foundations

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-20260930-r06-permutation-combination-foundations · 2026-09-30 · 100 questions

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



## 1. Exact Ordered Selection Count

Select k=0 objects in order, without replacement or repetition, from n=0 distinct objects. The empty selection counts once. Find P(0,0).

View answer and explanation1

Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0001/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 2. Exact Ordered Selection Count

Select k=1 objects in order, without replacement or repetition, from n=1 distinct objects. The empty selection counts once. Find P(1,1).

View answer and explanation1

Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0002/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 3. Exact Ordered Selection Count

Select k=0 objects in order, without replacement or repetition, from n=2 distinct objects. The empty selection counts once. Find P(2,0).

View answer and explanation1

Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0003/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 4. Exact Ordered Selection Count

Select k=2 objects in order, without replacement or repetition, from n=2 distinct objects. The empty selection counts once. Find P(2,2).

View answer and explanation2

Multiply the successive available-choice counts exactly to obtain 2; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0004/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 5. Exact Ordered Selection Count

Select k=1 objects in order, without replacement or repetition, from n=3 distinct objects. The empty selection counts once. Find P(3,1).

View answer and explanation3

Multiply the successive available-choice counts exactly to obtain 3; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0005/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 6. Exact Ordered Selection Count

Select k=3 objects in order, without replacement or repetition, from n=3 distinct objects. The empty selection counts once. Find P(3,3).

View answer and explanation6

Multiply the successive available-choice counts exactly to obtain 6; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0006/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 7. Exact Ordered Selection Count

Select k=2 objects in order, without replacement or repetition, from n=4 distinct objects. The empty selection counts once. Find P(4,2).

View answer and explanation12

Multiply the successive available-choice counts exactly to obtain 12; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0007/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 8. Exact Ordered Selection Count

Select k=4 objects in order, without replacement or repetition, from n=4 distinct objects. The empty selection counts once. Find P(4,4).

View answer and explanation24

Multiply the successive available-choice counts exactly to obtain 24; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0008/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 9. Exact Ordered Selection Count

Select k=0 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,0).

View answer and explanation1

Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0009/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 10. Exact Ordered Selection Count

Select k=1 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,1).

View answer and explanation5

Multiply the successive available-choice counts exactly to obtain 5; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0010/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 11. Exact Ordered Selection Count

Select k=2 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,2).

View answer and explanation20

Multiply the successive available-choice counts exactly to obtain 20; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0011/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 12. Exact Ordered Selection Count

Select k=5 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,5).

View answer and explanation120

Multiply the successive available-choice counts exactly to obtain 120; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0012/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 13. Exact Ordered Selection Count

Select k=1 objects in order, without replacement or repetition, from n=6 distinct objects. The empty selection counts once. Find P(6,1).

View answer and explanation6

Multiply the successive available-choice counts exactly to obtain 6; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0013/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 14. Exact Ordered Selection Count

Select k=3 objects in order, without replacement or repetition, from n=6 distinct objects. The empty selection counts once. Find P(6,3).

View answer and explanation120

Multiply the successive available-choice counts exactly to obtain 120; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0014/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 15. Exact Ordered Selection Count

Select k=6 objects in order, without replacement or repetition, from n=6 distinct objects. The empty selection counts once. Find P(6,6).

View answer and explanation720

Multiply the successive available-choice counts exactly to obtain 720; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0015/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 16. Exact Ordered Selection Count

Select k=2 objects in order, without replacement or repetition, from n=7 distinct objects. The empty selection counts once. Find P(7,2).

View answer and explanation42

Multiply the successive available-choice counts exactly to obtain 42; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0016/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 17. Exact Ordered Selection Count

Select k=4 objects in order, without replacement or repetition, from n=7 distinct objects. The empty selection counts once. Find P(7,4).

View answer and explanation840

Multiply the successive available-choice counts exactly to obtain 840; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0017/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 18. Exact Ordered Selection Count

Select k=7 objects in order, without replacement or repetition, from n=7 distinct objects. The empty selection counts once. Find P(7,7).

View answer and explanation5040

Multiply the successive available-choice counts exactly to obtain 5040; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0018/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 19. Exact Ordered Selection Count

Select k=0 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(8,0).

View answer and explanation1

Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0019/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 20. Exact Ordered Selection Count

Select k=3 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(8,3).

View answer and explanation336

Multiply the successive available-choice counts exactly to obtain 336; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0020/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 21. Exact Ordered Selection Count

Select k=8 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(8,8).

View answer and explanation40320

Multiply the successive available-choice counts exactly to obtain 40320; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0021/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 22. Exact Ordered Selection Count

Select k=1 objects in order, without replacement or repetition, from n=9 distinct objects. The empty selection counts once. Find P(9,1).

View answer and explanation9

Multiply the successive available-choice counts exactly to obtain 9; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0022/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 23. Exact Ordered Selection Count

Select k=4 objects in order, without replacement or repetition, from n=9 distinct objects. The empty selection counts once. Find P(9,4).

View answer and explanation3024

Multiply the successive available-choice counts exactly to obtain 3024; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0023/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 24. Exact Ordered Selection Count

Select k=5 objects in order, without replacement or repetition, from n=10 distinct objects. The empty selection counts once. Find P(10,5).

View answer and explanation30240

Multiply the successive available-choice counts exactly to obtain 30240; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0024/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 25. Exact Ordered Selection Count

Select k=6 objects in order, without replacement or repetition, from n=12 distinct objects. The empty selection counts once. Find P(12,6).

View answer and explanation665280

Multiply the successive available-choice counts exactly to obtain 665280; when k=0 the empty product is 1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-exact-count-0025/)

Permutations and combinations · qf-permutation-exact-count · candidate





## 26. Exact Unordered Selection Count

Select k=0 objects without order, replacement, or repetition from n=0 distinct objects. The empty selection counts once. Find C(0,0).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0001/)

Permutations and combinations · qf-combination-exact-count · candidate





## 27. Exact Unordered Selection Count

Select k=1 objects without order, replacement, or repetition from n=1 distinct objects. The empty selection counts once. Find C(1,1).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0002/)

Permutations and combinations · qf-combination-exact-count · candidate





## 28. Exact Unordered Selection Count

Select k=0 objects without order, replacement, or repetition from n=2 distinct objects. The empty selection counts once. Find C(2,0).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0003/)

Permutations and combinations · qf-combination-exact-count · candidate





## 29. Exact Unordered Selection Count

Select k=2 objects without order, replacement, or repetition from n=2 distinct objects. The empty selection counts once. Find C(2,2).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0004/)

Permutations and combinations · qf-combination-exact-count · candidate





## 30. Exact Unordered Selection Count

Select k=1 objects without order, replacement, or repetition from n=3 distinct objects. The empty selection counts once. Find C(3,1).

View answer and explanation3

Count the k-element subsets exactly to obtain 3; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0005/)

Permutations and combinations · qf-combination-exact-count · candidate





## 31. Exact Unordered Selection Count

Select k=3 objects without order, replacement, or repetition from n=3 distinct objects. The empty selection counts once. Find C(3,3).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0006/)

Permutations and combinations · qf-combination-exact-count · candidate





## 32. Exact Unordered Selection Count

Select k=2 objects without order, replacement, or repetition from n=4 distinct objects. The empty selection counts once. Find C(4,2).

View answer and explanation6

Count the k-element subsets exactly to obtain 6; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0007/)

Permutations and combinations · qf-combination-exact-count · candidate





## 33. Exact Unordered Selection Count

Select k=4 objects without order, replacement, or repetition from n=4 distinct objects. The empty selection counts once. Find C(4,4).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0008/)

Permutations and combinations · qf-combination-exact-count · candidate





## 34. Exact Unordered Selection Count

Select k=0 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,0).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0009/)

Permutations and combinations · qf-combination-exact-count · candidate





## 35. Exact Unordered Selection Count

Select k=1 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,1).

View answer and explanation5

Count the k-element subsets exactly to obtain 5; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0010/)

Permutations and combinations · qf-combination-exact-count · candidate





## 36. Exact Unordered Selection Count

Select k=2 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,2).

View answer and explanation10

Count the k-element subsets exactly to obtain 10; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0011/)

Permutations and combinations · qf-combination-exact-count · candidate





## 37. Exact Unordered Selection Count

Select k=5 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,5).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0012/)

Permutations and combinations · qf-combination-exact-count · candidate





## 38. Exact Unordered Selection Count

Select k=1 objects without order, replacement, or repetition from n=6 distinct objects. The empty selection counts once. Find C(6,1).

View answer and explanation6

Count the k-element subsets exactly to obtain 6; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0013/)

Permutations and combinations · qf-combination-exact-count · candidate





## 39. Exact Unordered Selection Count

Select k=3 objects without order, replacement, or repetition from n=6 distinct objects. The empty selection counts once. Find C(6,3).

View answer and explanation20

Count the k-element subsets exactly to obtain 20; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0014/)

Permutations and combinations · qf-combination-exact-count · candidate





## 40. Exact Unordered Selection Count

Select k=6 objects without order, replacement, or repetition from n=6 distinct objects. The empty selection counts once. Find C(6,6).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0015/)

Permutations and combinations · qf-combination-exact-count · candidate





## 41. Exact Unordered Selection Count

Select k=2 objects without order, replacement, or repetition from n=7 distinct objects. The empty selection counts once. Find C(7,2).

View answer and explanation21

Count the k-element subsets exactly to obtain 21; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0016/)

Permutations and combinations · qf-combination-exact-count · candidate





## 42. Exact Unordered Selection Count

Select k=4 objects without order, replacement, or repetition from n=7 distinct objects. The empty selection counts once. Find C(7,4).

View answer and explanation35

Count the k-element subsets exactly to obtain 35; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0017/)

Permutations and combinations · qf-combination-exact-count · candidate





## 43. Exact Unordered Selection Count

Select k=7 objects without order, replacement, or repetition from n=7 distinct objects. The empty selection counts once. Find C(7,7).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0018/)

Permutations and combinations · qf-combination-exact-count · candidate





## 44. Exact Unordered Selection Count

Select k=0 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,0).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0019/)

Permutations and combinations · qf-combination-exact-count · candidate





## 45. Exact Unordered Selection Count

Select k=3 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,3).

View answer and explanation56

Count the k-element subsets exactly to obtain 56; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0020/)

Permutations and combinations · qf-combination-exact-count · candidate





## 46. Exact Unordered Selection Count

Select k=8 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,8).

View answer and explanation1

Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0021/)

Permutations and combinations · qf-combination-exact-count · candidate





## 47. Exact Unordered Selection Count

Select k=1 objects without order, replacement, or repetition from n=9 distinct objects. The empty selection counts once. Find C(9,1).

View answer and explanation9

Count the k-element subsets exactly to obtain 9; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0022/)

Permutations and combinations · qf-combination-exact-count · candidate





## 48. Exact Unordered Selection Count

Select k=4 objects without order, replacement, or repetition from n=9 distinct objects. The empty selection counts once. Find C(9,4).

View answer and explanation126

Count the k-element subsets exactly to obtain 126; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0023/)

Permutations and combinations · qf-combination-exact-count · candidate





## 49. Exact Unordered Selection Count

Select k=5 objects without order, replacement, or repetition from n=10 distinct objects. The empty selection counts once. Find C(10,5).

View answer and explanation252

Count the k-element subsets exactly to obtain 252; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0024/)

Permutations and combinations · qf-combination-exact-count · candidate





## 50. Exact Unordered Selection Count

Select k=6 objects without order, replacement, or repetition from n=12 distinct objects. The empty selection counts once. Find C(12,6).

View answer and explanation924

Count the k-element subsets exactly to obtain 924; the empty subset is one valid selection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-combination-exact-count-0025/)

Permutations and combinations · qf-combination-exact-count · candidate





## 51. Choosing a Selection Model

Select k=0 objects with order distinguished, without replacement or repetition, from n=0 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0001/)

Permutations and combinations · qf-selection-model-choice · candidate





## 52. Choosing a Selection Model

Select k=1 objects with order ignored, without replacement or repetition, from n=1 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0002/)

Permutations and combinations · qf-selection-model-choice · candidate





## 53. Choosing a Selection Model

Select k=0 objects with order distinguished, without replacement or repetition, from n=2 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0003/)

Permutations and combinations · qf-selection-model-choice · candidate





## 54. Choosing a Selection Model

Select k=2 objects with order ignored, without replacement or repetition, from n=2 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0004/)

Permutations and combinations · qf-selection-model-choice · candidate





## 55. Choosing a Selection Model

Select k=1 objects with order distinguished, without replacement or repetition, from n=3 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0005/)

Permutations and combinations · qf-selection-model-choice · candidate





## 56. Choosing a Selection Model

Select k=3 objects with order ignored, without replacement or repetition, from n=3 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0006/)

Permutations and combinations · qf-selection-model-choice · candidate





## 57. Choosing a Selection Model

Select k=2 objects with order distinguished, without replacement or repetition, from n=4 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0007/)

Permutations and combinations · qf-selection-model-choice · candidate





## 58. Choosing a Selection Model

Select k=4 objects with order ignored, without replacement or repetition, from n=4 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0008/)

Permutations and combinations · qf-selection-model-choice · candidate





## 59. Choosing a Selection Model

Select k=0 objects with order distinguished, without replacement or repetition, from n=5 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0009/)

Permutations and combinations · qf-selection-model-choice · candidate





## 60. Choosing a Selection Model

Select k=1 objects with order ignored, without replacement or repetition, from n=5 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0010/)

Permutations and combinations · qf-selection-model-choice · candidate





## 61. Choosing a Selection Model

Select k=2 objects with order distinguished, without replacement or repetition, from n=5 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0011/)

Permutations and combinations · qf-selection-model-choice · candidate





## 62. Choosing a Selection Model

Select k=5 objects with order ignored, without replacement or repetition, from n=5 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0012/)

Permutations and combinations · qf-selection-model-choice · candidate





## 63. Choosing a Selection Model

Select k=1 objects with order distinguished, without replacement or repetition, from n=6 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0013/)

Permutations and combinations · qf-selection-model-choice · candidate





## 64. Choosing a Selection Model

Select k=3 objects with order ignored, without replacement or repetition, from n=6 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0014/)

Permutations and combinations · qf-selection-model-choice · candidate





## 65. Choosing a Selection Model

Select k=6 objects with order distinguished, without replacement or repetition, from n=6 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0015/)

Permutations and combinations · qf-selection-model-choice · candidate





## 66. Choosing a Selection Model

Select k=2 objects with order ignored, without replacement or repetition, from n=7 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0016/)

Permutations and combinations · qf-selection-model-choice · candidate





## 67. Choosing a Selection Model

Select k=4 objects with order distinguished, without replacement or repetition, from n=7 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0017/)

Permutations and combinations · qf-selection-model-choice · candidate





## 68. Choosing a Selection Model

Select k=7 objects with order ignored, without replacement or repetition, from n=7 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0018/)

Permutations and combinations · qf-selection-model-choice · candidate





## 69. Choosing a Selection Model

Select k=0 objects with order distinguished, without replacement or repetition, from n=8 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0019/)

Permutations and combinations · qf-selection-model-choice · candidate





## 70. Choosing a Selection Model

Select k=3 objects with order ignored, without replacement or repetition, from n=8 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0020/)

Permutations and combinations · qf-selection-model-choice · candidate





## 71. Choosing a Selection Model

Select k=8 objects with order distinguished, without replacement or repetition, from n=8 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0021/)

Permutations and combinations · qf-selection-model-choice · candidate





## 72. Choosing a Selection Model

Select k=1 objects with order ignored, without replacement or repetition, from n=9 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0022/)

Permutations and combinations · qf-selection-model-choice · candidate





## 73. Choosing a Selection Model

Select k=4 objects with order distinguished, without replacement or repetition, from n=9 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0023/)

Permutations and combinations · qf-selection-model-choice · candidate





## 74. Choosing a Selection Model

Select k=5 objects with order ignored, without replacement or repetition, from n=10 distinct objects. Should the model be a permutation or a combination?

View answer and explanationcombination

The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0024/)

Permutations and combinations · qf-selection-model-choice · candidate





## 75. Choosing a Selection Model

Select k=6 objects with order distinguished, without replacement or repetition, from n=12 distinct objects. Should the model be a permutation or a combination?

View answer and explanationpermutation

The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-selection-model-choice-0025/)

Permutations and combinations · qf-selection-model-choice · candidate





## 76. Quotient of Permutation and Combination Counts

For selection of k=0 distinct objects without replacement from n=0, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation1

Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0001/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 77. Quotient of Permutation and Combination Counts

For selection of k=1 distinct objects without replacement from n=1, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation1

Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0002/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 78. Quotient of Permutation and Combination Counts

For selection of k=0 distinct objects without replacement from n=2, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation1

Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0003/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 79. Quotient of Permutation and Combination Counts

For selection of k=2 distinct objects without replacement from n=2, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation2

Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0004/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 80. Quotient of Permutation and Combination Counts

For selection of k=1 distinct objects without replacement from n=3, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation1

Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0005/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 81. Quotient of Permutation and Combination Counts

For selection of k=3 distinct objects without replacement from n=3, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation6

Each k-element subset has k! orderings, so the quotient is 6; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0006/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 82. Quotient of Permutation and Combination Counts

For selection of k=2 distinct objects without replacement from n=4, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation2

Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0007/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 83. Quotient of Permutation and Combination Counts

For selection of k=4 distinct objects without replacement from n=4, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation24

Each k-element subset has k! orderings, so the quotient is 24; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0008/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 84. Quotient of Permutation and Combination Counts

For selection of k=0 distinct objects without replacement from n=5, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation1

Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0009/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 85. Quotient of Permutation and Combination Counts

For selection of k=1 distinct objects without replacement from n=5, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation1

Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0010/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 86. Quotient of Permutation and Combination Counts

For selection of k=2 distinct objects without replacement from n=5, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation2

Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0011/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 87. Quotient of Permutation and Combination Counts

For selection of k=5 distinct objects without replacement from n=5, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation120

Each k-element subset has k! orderings, so the quotient is 120; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0012/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 88. Quotient of Permutation and Combination Counts

For selection of k=1 distinct objects without replacement from n=6, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation1

Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0013/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 89. Quotient of Permutation and Combination Counts

For selection of k=3 distinct objects without replacement from n=6, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation6

Each k-element subset has k! orderings, so the quotient is 6; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0014/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 90. Quotient of Permutation and Combination Counts

For selection of k=6 distinct objects without replacement from n=6, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation720

Each k-element subset has k! orderings, so the quotient is 720; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0015/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 91. Quotient of Permutation and Combination Counts

For selection of k=2 distinct objects without replacement from n=7, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation2

Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0016/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 92. Quotient of Permutation and Combination Counts

For selection of k=4 distinct objects without replacement from n=7, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation24

Each k-element subset has k! orderings, so the quotient is 24; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0017/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 93. Quotient of Permutation and Combination Counts

For selection of k=7 distinct objects without replacement from n=7, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation5040

Each k-element subset has k! orderings, so the quotient is 5040; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0018/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 94. Quotient of Permutation and Combination Counts

For selection of k=0 distinct objects without replacement from n=8, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation1

Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0019/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 95. Quotient of Permutation and Combination Counts

For selection of k=3 distinct objects without replacement from n=8, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation6

Each k-element subset has k! orderings, so the quotient is 6; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0020/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 96. Quotient of Permutation and Combination Counts

For selection of k=8 distinct objects without replacement from n=8, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation40320

Each k-element subset has k! orderings, so the quotient is 40320; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0021/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 97. Quotient of Permutation and Combination Counts

For selection of k=1 distinct objects without replacement from n=9, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation1

Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0022/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 98. Quotient of Permutation and Combination Counts

For selection of k=4 distinct objects without replacement from n=9, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation24

Each k-element subset has k! orderings, so the quotient is 24; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0023/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 99. Quotient of Permutation and Combination Counts

For selection of k=5 distinct objects without replacement from n=10, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation120

Each k-element subset has k! orderings, so the quotient is 120; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0024/)

Permutations and combinations · qf-permutation-combination-quotient · candidate





## 100. Quotient of Permutation and Combination Counts

For selection of k=6 distinct objects without replacement from n=12, compute the exact integer quotient P(n,k)/C(n,k). The empty selection counts once.

View answer and explanation720

Each k-element subset has k! orderings, so the quotient is 720; the empty subset also has one ordering.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r06-permutation-combination-quotient-0025/)

Permutations and combinations · qf-permutation-combination-quotient · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260930-r06-permutation-combination-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260930-r06-permutation-combination-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260930-r06-permutation-combination-foundations/validation-report.json)