Combinatorial interpretation of multinomial function.

Question

Given $n$ items if we pick $k$ we use binomial function. What is the analogy with multinomial function?

Answer 1

Given boxes with sizes $k_1,\ldots,k_r$, with total size $n$, how many ways are there to place balls labeled $1,\ldots, n$ into those boxes?

Note that if $r=2$, you are choosing $k$ balls to go into the first box, and the rest go into the second box.

To see this, look at the definition, $$\binom{n}{k_1,\ldots,k_r} = \frac{n!}{k_1!\cdots k_r!}.$$

We arrange the balls in some order, then the first $k_1$ go in the first box, the next $k_2$ go in the second box, and so on. But then we have overcounted by the number of ways to arrange the balls within each box, which total $k_1!\cdots k_r!$, so we divide by this to get the multinomial coefficient.

For more information, see the part of the wikipedia page on interpretations of the multinomial coefficient.