Combinatorics Calculator

Compute factorials, permutations, combinations, and multinomial coefficients. See step-by-step formula breakdowns and explore Pascal's Triangle interactively. All calculations run in your browser.

Result

5! = 120

Step-by-step breakdown

5! = 5 \times 4 \times 3 \times 2 \times 1 = 120

Pascal's Triangle

Click any cell to compute that combination C(n, r).

Reference Guide

Factorials

The factorial of n counts the number of ways to arrange n distinct objects in a line.

n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1

By convention, $0! = 1$ . Factorials grow very fast. For example, $10! = 3{,}628{,}800$ and $20!$ is already a 19-digit number.

Permutations

A permutation counts the number of ordered arrangements of r items chosen from n distinct items. Order matters.

P(n, r) = \frac{n!}{(n-r)!}

For example, the number of ways to award gold, silver, and bronze medals to 3 of 10 athletes is $P(10, 3) = 720$ .

Combinations

A combination counts the number of ways to choose r items from n distinct items when order does not matter.

C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Combinations appear as entries in Pascal's Triangle, where row n and column r give $C(n, r)$ . A useful identity is $C(n, r) = C(n, n-r)$ .

Multinomial Coefficients

The multinomial coefficient counts the number of ways to divide n objects into groups of sizes $k_1, k_2, \ldots, k_m$ where $k_1 + k_2 + \cdots + k_m = n$ .

\binom{n}{k_1, k_2, \ldots, k_m} = \frac{n!}{k_1! \cdot k_2! \cdots k_m!}

A classic example is counting arrangements of the letters in MISSISSIPPI. With $n = 11$ letters and groups $(1, 4, 4, 2)$ for M, I, S, P, the answer is $\frac{11!}{1! \cdot 4! \cdot 4! \cdot 2!} = 34{,}650$ .