Multinomial coefficient

Definition

Let $k_1, \dots, k_m$ be natural numbers and let $n = k_1 + \dots + k_m$. The multinomial coefficient is the natural number $$\binom{n}{k_1, k_2, \dots, k_m} = \frac{n!}{k_1!\, k_2! \cdots k_m!}.$$

Remarks

For $m = 2$ it is the binomial coefficient: $\binom{n}{k, n-k} = \binom{n}{k}$. It counts the ways to put $n$ distinct objects into $m$ labelled boxes, with $k_i$ objects in box $i$. It also counts the distinct words that use a letter $a_i$ exactly $k_i$ times. For example, MISSISSIPPI has $\frac{11!}{1!\,4!\,4!\,2!} = 34650$ distinct rearrangements. It is the coefficient in the multinomial theorem.

Explanations

L1 human

The multinomial coefficient counts the ways to split $n$ objects into groups of given sizes. To split 4 people into a group of 2 and two groups of 1, with labelled groups, there are $\frac{4!}{2!\,1!\,1!} = 12$ ways.