Multinomial Theorem - Multinomial Coefficients

Multinomial Coefficients

The numbers

${n \choose k_1, k_2, \ldots, k_m} = \frac{n!}{k_1!\, k_2! \cdots k_m!}$

(which can also be written as:)

$= {k_1\choose k_1}{k_1+k_2\choose k_2}\cdots{k_1+k_2+\cdots+k_m\choose k_m} = \prod_{i=1}^m {\sum_{j=1}^i k_j \choose k_i}$

are the multinomial coefficients. Just like "n choose k" are the coefficients when you raise a binomial to the nth power (e.g. the coefficients are 1,3,3,1 for (a + b)3, where n = 3), the multinomial coefficients appear when one raises a multinomial to the nth power (e.g. (a + b + c)3)

Read more about this topic: Multinomial Theorem