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.
Select k=0 objects in order, without replacement or repetition, from n=0 distinct objects. The empty selection counts once. Find P(0,0).
1
Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=1 objects in order, without replacement or repetition, from n=1 distinct objects. The empty selection counts once. Find P(1,1).
1
Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=0 objects in order, without replacement or repetition, from n=2 distinct objects. The empty selection counts once. Find P(2,0).
1
Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=2 objects in order, without replacement or repetition, from n=2 distinct objects. The empty selection counts once. Find P(2,2).
2
Multiply the successive available-choice counts exactly to obtain 2; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=1 objects in order, without replacement or repetition, from n=3 distinct objects. The empty selection counts once. Find P(3,1).
3
Multiply the successive available-choice counts exactly to obtain 3; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=3 objects in order, without replacement or repetition, from n=3 distinct objects. The empty selection counts once. Find P(3,3).
6
Multiply the successive available-choice counts exactly to obtain 6; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=2 objects in order, without replacement or repetition, from n=4 distinct objects. The empty selection counts once. Find P(4,2).
12
Multiply the successive available-choice counts exactly to obtain 12; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=4 objects in order, without replacement or repetition, from n=4 distinct objects. The empty selection counts once. Find P(4,4).
24
Multiply the successive available-choice counts exactly to obtain 24; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=0 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,0).
1
Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=1 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,1).
5
Multiply the successive available-choice counts exactly to obtain 5; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=2 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,2).
20
Multiply the successive available-choice counts exactly to obtain 20; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=5 objects in order, without replacement or repetition, from n=5 distinct objects. The empty selection counts once. Find P(5,5).
120
Multiply the successive available-choice counts exactly to obtain 120; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=1 objects in order, without replacement or repetition, from n=6 distinct objects. The empty selection counts once. Find P(6,1).
6
Multiply the successive available-choice counts exactly to obtain 6; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=3 objects in order, without replacement or repetition, from n=6 distinct objects. The empty selection counts once. Find P(6,3).
120
Multiply the successive available-choice counts exactly to obtain 120; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=6 objects in order, without replacement or repetition, from n=6 distinct objects. The empty selection counts once. Find P(6,6).
720
Multiply the successive available-choice counts exactly to obtain 720; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=2 objects in order, without replacement or repetition, from n=7 distinct objects. The empty selection counts once. Find P(7,2).
42
Multiply the successive available-choice counts exactly to obtain 42; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=4 objects in order, without replacement or repetition, from n=7 distinct objects. The empty selection counts once. Find P(7,4).
840
Multiply the successive available-choice counts exactly to obtain 840; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=7 objects in order, without replacement or repetition, from n=7 distinct objects. The empty selection counts once. Find P(7,7).
5040
Multiply the successive available-choice counts exactly to obtain 5040; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=0 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(8,0).
1
Multiply the successive available-choice counts exactly to obtain 1; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=3 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(8,3).
336
Multiply the successive available-choice counts exactly to obtain 336; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=8 objects in order, without replacement or repetition, from n=8 distinct objects. The empty selection counts once. Find P(8,8).
40320
Multiply the successive available-choice counts exactly to obtain 40320; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=1 objects in order, without replacement or repetition, from n=9 distinct objects. The empty selection counts once. Find P(9,1).
9
Multiply the successive available-choice counts exactly to obtain 9; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=4 objects in order, without replacement or repetition, from n=9 distinct objects. The empty selection counts once. Find P(9,4).
3024
Multiply the successive available-choice counts exactly to obtain 3024; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=5 objects in order, without replacement or repetition, from n=10 distinct objects. The empty selection counts once. Find P(10,5).
30240
Multiply the successive available-choice counts exactly to obtain 30240; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=6 objects in order, without replacement or repetition, from n=12 distinct objects. The empty selection counts once. Find P(12,6).
665280
Multiply the successive available-choice counts exactly to obtain 665280; when k=0 the empty product is 1.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-exact-count · candidate
Select k=0 objects without order, replacement, or repetition from n=0 distinct objects. The empty selection counts once. Find C(0,0).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=1 objects without order, replacement, or repetition from n=1 distinct objects. The empty selection counts once. Find C(1,1).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=0 objects without order, replacement, or repetition from n=2 distinct objects. The empty selection counts once. Find C(2,0).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=2 objects without order, replacement, or repetition from n=2 distinct objects. The empty selection counts once. Find C(2,2).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=1 objects without order, replacement, or repetition from n=3 distinct objects. The empty selection counts once. Find C(3,1).
3
Count the k-element subsets exactly to obtain 3; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=3 objects without order, replacement, or repetition from n=3 distinct objects. The empty selection counts once. Find C(3,3).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=2 objects without order, replacement, or repetition from n=4 distinct objects. The empty selection counts once. Find C(4,2).
6
Count the k-element subsets exactly to obtain 6; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=4 objects without order, replacement, or repetition from n=4 distinct objects. The empty selection counts once. Find C(4,4).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=0 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,0).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=1 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,1).
5
Count the k-element subsets exactly to obtain 5; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=2 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,2).
10
Count the k-element subsets exactly to obtain 10; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=5 objects without order, replacement, or repetition from n=5 distinct objects. The empty selection counts once. Find C(5,5).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=1 objects without order, replacement, or repetition from n=6 distinct objects. The empty selection counts once. Find C(6,1).
6
Count the k-element subsets exactly to obtain 6; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=3 objects without order, replacement, or repetition from n=6 distinct objects. The empty selection counts once. Find C(6,3).
20
Count the k-element subsets exactly to obtain 20; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=6 objects without order, replacement, or repetition from n=6 distinct objects. The empty selection counts once. Find C(6,6).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=2 objects without order, replacement, or repetition from n=7 distinct objects. The empty selection counts once. Find C(7,2).
21
Count the k-element subsets exactly to obtain 21; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=4 objects without order, replacement, or repetition from n=7 distinct objects. The empty selection counts once. Find C(7,4).
35
Count the k-element subsets exactly to obtain 35; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=7 objects without order, replacement, or repetition from n=7 distinct objects. The empty selection counts once. Find C(7,7).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=0 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,0).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=3 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,3).
56
Count the k-element subsets exactly to obtain 56; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=8 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,8).
1
Count the k-element subsets exactly to obtain 1; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=1 objects without order, replacement, or repetition from n=9 distinct objects. The empty selection counts once. Find C(9,1).
9
Count the k-element subsets exactly to obtain 9; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=4 objects without order, replacement, or repetition from n=9 distinct objects. The empty selection counts once. Find C(9,4).
126
Count the k-element subsets exactly to obtain 126; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=5 objects without order, replacement, or repetition from n=10 distinct objects. The empty selection counts once. Find C(10,5).
252
Count the k-element subsets exactly to obtain 252; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
Select k=6 objects without order, replacement, or repetition from n=12 distinct objects. The empty selection counts once. Find C(12,6).
924
Count the k-element subsets exactly to obtain 924; the empty subset is one valid selection.
Reviewed step-by-step derivation
Permutations and combinations · qf-combination-exact-count · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
combination
The objects are distinct and selection is without replacement. Because order is ignored, use the combination model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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?
permutation
The objects are distinct and selection is without replacement. Because order is significant, use the permutation model.
Reviewed step-by-step derivation
Permutations and combinations · qf-selection-model-choice · candidate
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.
1
Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
1
Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
1
Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
2
Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
1
Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
6
Each k-element subset has k! orderings, so the quotient is 6; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
2
Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
24
Each k-element subset has k! orderings, so the quotient is 24; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
1
Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
1
Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
2
Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
120
Each k-element subset has k! orderings, so the quotient is 120; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
1
Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
6
Each k-element subset has k! orderings, so the quotient is 6; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
720
Each k-element subset has k! orderings, so the quotient is 720; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
2
Each k-element subset has k! orderings, so the quotient is 2; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
24
Each k-element subset has k! orderings, so the quotient is 24; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
5040
Each k-element subset has k! orderings, so the quotient is 5040; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
1
Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
6
Each k-element subset has k! orderings, so the quotient is 6; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
40320
Each k-element subset has k! orderings, so the quotient is 40320; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
1
Each k-element subset has k! orderings, so the quotient is 1; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
24
Each k-element subset has k! orderings, so the quotient is 24; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
120
Each k-element subset has k! orderings, so the quotient is 120; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
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.
720
Each k-element subset has k! orderings, so the quotient is 720; the empty subset also has one ordering.
Reviewed step-by-step derivation
Permutations and combinations · qf-permutation-combination-quotient · candidate
Canonical question JSONL · English JSONL · Validation report