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Toolbox · Permutations and binomials

Permutations and binomials

Permutations, combinations, factorials and binomial expansion: coefficient table, general term and Pascal triangle, all in exact big integers.

Formula

A_n^m = \frac{n!}{(n-m)!},\quad C_n^m = \frac{n!}{m!\,(n-m)!}

A permutation multiplies m descending factors from n; a combination divides that by m factorial.

ModeTotal nChosen m

n goes up to 60, or 20 for a binomial expansion; m cannot exceed n.

Presets

Combinations120

n10

Total n10

Chosen m3

Combinations120

Pascal triangle (the highlighted row is the expansion)

121

1331

14641

15101051

1615201561

172135352171

18285670562881

Symmetry: C(10, 7) = 120
Permutations over m factorial: A(10, 3) / 3! = 720 / 6
Row total: 1 024

Steps

4 / 4

1Formula
$C_{10}^{3} = \frac{10!}{3!\,(10-3)!}$
A combination is a permutation divided by m factorial.
2Substitute
$C_{10}^{3} = C_{10}^{7}$
Choosing m of n is the same as leaving n−m behind.
3Solve
$C_{10}^{3} = 120$
Substitute the numbers.
4Conclusion
$\sum_{k=0}^{10} C_{10}^{k} = 2^{10} = 1 024$
A whole row sums to 2 to the n.

Concept

A permutation cares about order and a combination does not. Picking two people to queue is a permutation; picking two to sweep is a combination.

Entry k of row n in Pascal triangle is exactly C(n,k), and each entry is the sum of the two above it.

Going further

Properties of binomial coefficients: Each row is symmetric and peaks in the middle, its total is 2 to the n, and the odd and even entries each sum to 2 to the n−1.

Common mistakes

Ask first whether order matters: yes means permutation, no means combination.
Cancel before multiplying so the intermediate numbers stay small.

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