The multinomial coefficient, like the binomial coefficient, has several combinatorial interpretations.
Let be nonnegative integers, and let . The multinomial coefficient is:
(1) the number of ways to put interchangeable objects into boxes, so that box has objects in it, for .
(2) the number of ways to choose interchangeable objects from objects, then to choose from what remains, then to choose from what remains, ..., then to choose from what remains.
(3) the number of unique permutations of a word with letters and distinct letters, such that the th letter occurs times.
(4) the product
(5) the quotient
(1) and (2) are clearly equivalent, and (2) and (4) are equivalent from the definition of the binomial coefficient. (4) and (5) are equivalent by simple algebra. There are a few ways to see that (3) is equivalent to the others. Arguing combinatorially, note that a permutation of a word as in (3) corresponds to choices of spots to put each of the repeated letters in; out of the spots , choose of those spots to put the first letter in, then spots out of the remaining to put the second letter in, and so on. So (3) is equivalent to (2).
(One can also count permutations directly, by taking permutations and dividing by factors that account for duplicates: divide by a factor of to account for the fact that permuting all of the first letters doesn't change the permutation, divide by to do the same for the second letters, and so on, which gives the formula from (5).)
How many unique permutations of the word ABRACADABRA are there?
Solution: Using definition (3), this is the multinomial coefficient
For a fixed and , what is the sum of all the multinomial coefficients ?
Using definition (1), this is the number of ways to put objects into boxes (each specific multinomial coefficient gives a different breakdown of how many objects are in each box). Each object has choices for its destination, so the total number is .
This example has a different solution using the multinomial theorem; see that wiki for details.
The multinomial coefficients are the coefficients of the terms in the expansion of ; in particular, the coefficient of is . This is the multinomial theorem.