In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials.
Theorem
For any positive integer and any non-negative integer , the multinomial theorem describes how a sum with terms expands when raised to the th power:
<math display="block">(x_1 + x_2 + \cdots + x_m)^n
= \sum_{\begin{array}{c} k_1+k_2+\cdots+k_m=n \\ k_1, k_2, \cdots, k_m \geq 0\end{array {n \choose k_1, k_2, \ldots, k_m}
x_1^{k_1} \cdot x_2^{k_2} \cdots x_m^{k_m}</math>
where
<math display="block"> {n \choose k_1, k_2, \ldots, k_m}
= \frac{n!}{k_1!\, k_2! \cdots k_m!}</math>
is a multinomial coefficient. The sum is taken over all combinations of nonnegative integer indices through such that the sum of all is . That is, for each term in the expansion, the exponents of the must add up to .
In the case , this statement reduces to that of the binomial theorem.
Number of ways to select according to a distribution
In statistical mechanics and combinatorics, if one has a number distribution of labels, then the multinomial coefficients naturally arise from the binomial coefficients. Given a number distribution on a set of total items, represents the number of items to be given the label . (In statistical mechanics is the label of the energy state.)
The number of arrangements is found by
- Choosing of the total to be labeled 1. This can be done <math>\tbinom{N}{n_1}</math> ways.
- From the remaining items choose to label 2. This can be done <math>\tbinom{N-n_1}{n_2}</math> ways.
- From the remaining items choose to label 3. Again, this can be done <math>\tbinom{N-n_1-n_2}{n_3}</math> ways.
Multiplying the number of choices at each step results in:
:<math>{N \choose n_1}{N-n_1\choose n_2}{N-n_1-n_2\choose n_3}\cdots=\frac{N!}{(N-n_1)!n_1!} \cdot \frac{(N-n_1)!}{(N-n_1-n_2)!n_2!} \cdot \frac{(N-n_1-n_2)!}{(N-n_1-n_2-n_3)!n_3!}\cdots.</math>
Cancellation results in the formula given above.
Number of unique permutations of words
thumb|Multinomial coefficient as a product of binomial coefficients, counting the permutations of the letters of MISSISSIPPI.
The multinomial coefficient
:<math>\binom{n}{k_1, \ldots, k_m}</math>
is also the number of distinct ways to permute a multiset of elements, where is the multiplicity of each of the th element. For example, the number of distinct permutations of the letters of the word MISSISSIPPI, which has 1 M, 4 Is, 4 Ss, and 2 Ps, is
:<math>{11 \choose 1, 4, 4, 2} = \frac{11!}{1!\, 4!\, 4!\, 2!} = 34650.</math>
Generalized Pascal's triangle
One can use the multinomial theorem to generalize Pascal's triangle or Pascal's pyramid to Pascal's simplex. This provides a quick way to generate a lookup table for multinomial coefficients.
A related structure is the multinomial triangle, or generalized Pascal triangle of order m, which may be constructed using the recurrence relation:
<math display="block">\binom{n}{k}_{m-1} = \sum_{i=0}^{m-1} \binom{n-1}{k-i}_{m-1}</math>
from which Pascal's rule is recovered when <math>m=2</math>. These multinomial coefficients can be written as closed-form expressions with bounded integer compositions:
<math display="block">
\binom{n}{k}_{m-1} = \sum_{\begin{array}{c} k_0 + k_1 + \cdots + k_{m-1} = n \\
k_1 + 2k_2 + \cdots + (m-1)k_{m-1} = k\end{array {n \choose k_0, k_1, \ldots, k_{m-1
</math>
and without:
<math display="block">\binom{n}{k}_{m-1} = \sum_{i = 0}^{\lfloor k / m \rfloor} (-1)^i \binom{n}{i} \binom{n - 1 + k - im}{n-1}</math>
See also
- Multinomial distribution
- Stars and bars (combinatorics)