Combinatorics Calculator

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

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

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

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

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

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

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

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 $\frac{11!}{1! \cdot 4! \cdot 4! \cdot 2!} = 34{,}650$.