Multinomial theorem
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.
For any positive integer m and any non-negative integer n, the multinomial theorem describes how a sum with m terms expands when raised to the nth power:
{\displaystyle (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}}}
is a multinomial coefficient.
The sum is taken over all combinations of nonnegative integer indices k1 through km such that the sum of all ki is n. That is, for each term in the expansion, the exponents of the xi must add up to n.[1][a] In the case m = 2, this statement reduces to that of the binomial theorem.
[1] The third power of the trinomial a + b + c is given by
This can be computed by hand using the distributive property of multiplication over addition and combining like terms, but it can also be done (perhaps more easily) with the multinomial theorem.
It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula.
For example, the term
The statement of the theorem can be written concisely using multiindices: where and This proof of the multinomial theorem uses the binomial theorem and induction on m. First, for m = 1, both sides equal x1n since there is only one term k1 = n in the sum.
For the induction step, suppose the multinomial theorem holds for m. Then by the induction hypothesis.
Applying the binomial theorem to the last factor, which completes the induction.
The last step follows because as can easily be seen by writing the three coefficients using factorials as follows: The numbers appearing in the theorem are the multinomial coefficients.
They can be expressed in numerous ways, including as a product of binomial coefficients or of factorials: The substitution of xi = 1 for all i into the multinomial theorem gives immediately that The number of terms in a multinomial sum, #n,m, is equal to the number of monomials of degree n on the variables x1, …, xm: The count can be performed easily using the method of stars and bars.
The largest power of a prime p that divides a multinomial coefficient may be computed using a generalization of Kummer's theorem.
By Stirling's approximation, or equivalently the log-gamma function's asymptotic expansion,
log
= k n log ( k ) +
log ( k ) − ( k − 1 ) log ( 2 π n )
{\displaystyle \log {\binom {kn}{n,n,\cdots ,n}}=kn\log(k)+{\frac {1}{2}}\left(\log(k)-(k-1)\log(2\pi n)\right)-{\frac {k^{2}-1}{12kn}}+{\frac {k^{4}-1}{360k^{3}n^{3}}}-{\frac {k^{6}-1}{1260k^{5}n^{5}}}+O\left({\frac {1}{n^{6}}}\right)}
n π
The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing n distinct objects into m distinct bins, with k1 objects in the first bin, k2 objects in the second bin, and so on.
[2] 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 {ni} on a set of N total items, ni represents the number of items to be given the label i.
(In statistical mechanics i is the label of the energy state.)
The number of arrangements is found by Multiplying the number of choices at each step results in: Cancellation results in the formula given above.
The multinomial coefficient is also the number of distinct ways to permute a multiset of n elements, where ki is the multiplicity of each of the ith 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 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.
