The Multinomial Theorem — Mathematics Knowledge System

Overview

The multinomial theorem generalizes the binomial theorem from two terms to any number of terms. It expresses the expansion of (x₁+x₂+⋯+xₘ)ⁿ as a sum over all ways of distributing the exponent n among the m variables, weighted by multinomial coefficients that count the number of ways to arrange a multiset. It is the natural counting tool whenever objects are being placed into more than two categories, such as distributing a hand of cards among several suits or arranging a word with repeated letters.

Intuition

The multinomial coefficient n!/(k₁!k₂!⋯kₘ!) counts the number of distinct arrangements of a multiset with kᵢ copies of the i-th symbol — for instance, the number of distinguishable ways to arrange the letters of a word with repeated letters. The multinomial theorem simply says that expanding (x₁+⋯+xₘ)ⁿ and collecting like terms produces exactly these counts as coefficients, because each term x₁^{k₁}⋯xₘ^{kₘ} arises once for every way of picking which of the n factors contributes an x₁, which contributes an x₂, and so on.

Formal Definition

Definition

For a positive integer n and m variables, the multinomial theorem states:

(x_1+x_2+\cdots+x_m)^n = \sum_{k_1+k_2+\cdots+k_m=n} \binom{n}{k_1,k_2,\ldots,k_m} x_1^{k_1}x_2^{k_2}\cdots x_m^{k_m}

Multinomial theorem

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

Multinomial coefficient

Worked Examples

List all (k1,k2,k3) with k1+k2+k3=2 and compute each multinomial coefficient.
$\binom{2}{2,0,0}=1,\ \binom{2}{0,2,0}=1,\ \binom{2}{0,0,2}=1,\ \binom{2}{1,1,0}=2,\ \binom{2}{1,0,1}=2,\ \binom{2}{0,1,1}=2$
Assemble the expansion.
$(x+y+z)^2 = x^2+y^2+z^2+2xy+2xz+2yz$

Answer: x²+y²+z²+2xy+2xz+2yz.

Practice Problems

Difficulty 3/10

What is the coefficient of x²y z in the expansion of (x+y+z)⁴?

Difficulty 4/10

How many distinct arrangements are there of the letters in 'MISSISSIPPI' (11 letters: M=1, I=4, S=4, P=2)?

Difficulty 3/10

Compute the multinomial coefficient C(5; 3,2) (n=5 split into groups of size 3 and 2).

Quiz

The multinomial coefficient n!/(k1!k2!...km!) counts:

the number of subsets of size k1the number of distinct arrangements of a multiset with k1, k2, ..., km copies of each symbolthe number of ways to choose k1 items from nthe value of n choose k1 only

What is the coefficient of xy²z in the expansion of (x+y+z)⁴?

412246

Summary

The multinomial theorem generalizes the binomial theorem to expansions of sums with more than two terms.
Multinomial coefficients n!/(k1!⋯km!) count arrangements of a multiset and appear as the coefficients in the expansion.
When m=2 the multinomial theorem reduces exactly to the ordinary binomial theorem.

References

WebsiteWikipedia — Multinomial theorem