Multinomial theorem#Ways to put objects into bins

For any positive integer {{mvar|m}} and any non-negative integer {{mvar|n}}, the multinomial theorem describes how a sum with {{mvar|m}} terms expands when raised to the {{mvar|n}}th power:

$$(x\_1\; +\; x\_2\; +\; \backslash cdots\; +\; x\_m)^n$$

= \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}

where

$$\{n\; \backslash choose\; k\_1,\; k\_2,\; \backslash ldots,\; k\_m\}$$

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

is a multinomial coefficient.{{citation |first=Martin |last=Aigner |author-link=Martin Aigner |title=Combinatorial Theory |year=1997 |publisher=Springer |page=77 }} The sum is taken over all combinations of nonnegative integer indices {{math|k{{sub|1}}}} through {{mvar|k{{sub|m}}}} such that the sum of all {{mvar|k{{sub|i}}}} is {{mvar|n}}. That is, for each term in the expansion, the exponents of the {{mvar|x{{sub|i}}}} must add up to {{mvar|n}}.{{citation |last = Stanley | first = Richard | author-link = Richard P. Stanley | title = Enumerative Combinatorics | volume = 1 | edition = 2 | at = §1.2 | publisher = Cambridge University Press| year = 2012}}{{efn|As with the binomial theorem, quantities of the form {{math|x{{sup|0}}}} that appear are taken to equal 1, Zero to the power of zero.}}

In the case {{math|1=m = 2}}, this statement reduces to that of the binomial theorem.

The third power of the trinomial {{math|a + b + c}} is given by

(a+b+c)^3 = a^3 + b^3 + c^3 + 3 a^2 b + 3 a^2 c + 3 b^2 a + 3 b^2 c + 3 c^2 a + 3 c^2 b + 6 a b c.

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 $a^2\; b^0\; c^1$ has coefficient $\{3\; \backslash choose\; 2,\; 0,\; 1\}\; =\; \backslash frac\{3!\}\{2!\backslash cdot\; 0!\backslash cdot\; 1!\}\; =\; \backslash frac\{6\}\{2\; \backslash cdot\; 1\; \backslash cdot\; 1\}\; =\; 3$, the term $a^1\; b^1\; c^1$ has coefficient $\{3\; \backslash choose\; 1,\; 1,\; 1\}\; =\; \backslash frac\{3!\}\{1!\backslash cdot\; 1!\backslash cdot\; 1!\}\; =\; \backslash frac\{6\}\{1\; \backslash cdot\; 1\; \backslash cdot\; 1\}\; =\; 6$, and so on.

The statement of the theorem can be written concisely using multiindices:

:$(x\_1+\backslash cdots+x\_m)^n\; =\; \backslash sum\_\{|\backslash alpha|=n\}\{n\; \backslash choose\; \backslash alpha\}x^\backslash alpha$

where

:$$

\alpha=(\alpha_1,\alpha_2,\dots,\alpha_m)

and

:$$

x^\alpha=x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_m^{\alpha_m}

This proof of the multinomial theorem uses the binomial theorem and induction on {{mvar|m}}.

First, for {{math|1=m = 1}}, both sides equal {{math|x{{sub|1}}{{sup|n}}}} since there is only one term {{math|1=k{{sub|1}} = n}} in the sum. For the induction step, suppose the multinomial theorem holds for {{mvar|m}}. Then

: $$

\begin{align}

& (x_1+x_2+\cdots+x_m+x_{m+1})^n = (x_1+x_2+\cdots+(x_m+x_{m+1}))^n \\[6pt]

= {} & \sum_{k_1+k_2+\cdots+k_{m-1}+K=n}{n\choose k_1,k_2,\ldots,k_{m-1},K} x_1^{k_1} x_2^{k_2}\cdots x_{m-1}^{k_{m-1}}(x_m+x_{m+1})^K

\end{align}

by the induction hypothesis. Applying the binomial theorem to the last factor,

:$=\; \backslash sum\_\{k\_1+k\_2+\backslash cdots+k\_\{m-1\}+K=n\}\{n\backslash choose\; k\_1,k\_2,\backslash ldots,k\_\{m-1\},K\}\; x\_1^\{k\_1\}x\_2^\{k\_2\}\backslash cdots\; x\_\{m-1\}^\{k\_\{m-1\}\}\backslash sum\_\{k\_m+k\_\{m+1\}=K\}\{K\backslash choose\; k\_m,k\_\{m+1\}\}x\_m^\{k\_m\}x\_\{m+1\}^\{k\_\{m+1\}\}$

:$=\; \backslash sum\_\{k\_1+k\_2+\backslash cdots+k\_\{m-1\}+k\_m+k\_\{m+1\}=n\}\{n\backslash choose\; k\_1,k\_2,\backslash ldots,k\_\{m-1\},k\_m,k\_\{m+1\}\}\; x\_1^\{k\_1\}x\_2^\{k\_2\}\backslash cdots\; x\_\{m-1\}^\{k\_\{m-1\}\}x\_m^\{k\_m\}x\_\{m+1\}^\{k\_\{m+1\}\}$

which completes the induction. The last step follows because

:$\{n\backslash choose\; k\_1,k\_2,\backslash ldots,k\_\{m-1\},K\}\{K\backslash choose\; k\_m,k\_\{m+1\}\}\; =\; \{n\backslash choose\; k\_1,k\_2,\backslash ldots,k\_\{m-1\},k\_m,k\_\{m+1\}\},$

as can easily be seen by writing the three coefficients using factorials as follows:

:$\backslash frac\{n!\}\{k\_1!\; k\_2!\; \backslash cdots\; k\_\{m-1\}!K!\}\; \backslash frac\{K!\}\{k\_m!\; k\_\{m+1\}!\}=\backslash frac\{n!\}\{k\_1!\; k\_2!\; \backslash cdots\; k\_\{m+1\}!\}.$

The numbers

:$\{n\; \backslash choose\; k\_1,\; k\_2,\; \backslash ldots,\; k\_m\}$

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:

:$$

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

The substitution of {{math|1=x{{sub|i}} = 1}} for all {{mvar|i}} into the multinomial theorem

:$\backslash sum\_\{k\_1+k\_2+\backslash cdots+k\_m=n\}\; \{n\; \backslash choose\; k\_1,\; k\_2,\; \backslash ldots,\; k\_m\}\; x\_1^\{k\_1\}\; x\_2^\{k\_2\}\; \backslash cdots\; x\_m^\{k\_m\}$

= (x_1 + x_2 + \cdots + x_m)^n

gives immediately that

:$$

\sum_{k_1+k_2+\cdots+k_m=n} {n \choose k_1, k_2, \ldots, k_m} = m^n.

The number of terms in a multinomial sum, {{math|#{{sub|n,m}}}}, is equal to the number of monomials of degree {{mvar|n}} on the variables {{math|x{{sub|1}}, …, x{{sub|m}}}}:

:$$

\#_{n,m} = {n+m-1 \choose m-1}.

The count can be performed easily using the method of stars and bars.

The largest power of a prime {{mvar|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, $$\backslash log\backslash binom\{kn\}\{n,\; n,\; \backslash cdots,\; n\}\; =\; k\; n\; \backslash log(k)\; +\; \backslash frac\{1\}\{2\}\; \backslash left(\backslash log(k)\; -\; (k\; -\; 1)\; \backslash log(2\; \backslash pi\; n)\backslash right)\; -\; \backslash frac\{k^2\; -\; 1\}\{12kn\}\; +\; \backslash frac\{k^4\; -\; 1\}\{360k^3n^3\}\; -\; \backslash frac\{k^6\; -\; 1\}\{1260k^5n^5\}\; +\; O\backslash left(\backslash frac\{1\}\{n^6\}\backslash right)$$so for example,$$\backslash binom\{2n\}\{n\}\; \backslash sim\; \backslash frac\{2^\{2n\}\}\{\backslash sqrt\{n\backslash pi\; \}\}$$

The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing {{mvar|n}} distinct objects into {{mvar|m}} distinct bins, with {{math|k{{sub|1}}}} objects in the first bin, {{math|k{{sub|2}}}} objects in the second bin, and so on.{{cite web |url=http://dlmf.nist.gov/ |title=NIST Digital Library of Mathematical Functions |author=National Institute of Standards and Technology |author-link=National Institute of Standards and Technology |date=May 11, 2010 |at=[http://dlmf.nist.gov/26.4 Section 26.4] |accessdate=August 30, 2010}}

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 {{math|{n{{sub|i}}} }} on a set of {{mvar|N}} total items, {{mvar|n{{sub|i}}}} represents the number of items to be given the label {{mvar|i}}. (In statistical mechanics {{mvar|i}} is the label of the energy state.)

The number of arrangements is found by

• Choosing {{math|n{{sub|1}}}} of the total {{mvar|N}} to be labeled 1. This can be done $\backslash tbinom\{N\}\{n\_1\}$ ways.
• From the remaining {{math|N − n{{sub|1}}}} items choose {{math|n{{sub|2}}}} to label 2. This can be done $\backslash tbinom\{N-n\_1\}\{n\_2\}$ ways.
• From the remaining {{math|N − n{{sub|1}} − n{{sub|2}}}} items choose {{math|n{{sub|3}}}} to label 3. Again, this can be done $\backslash tbinom\{N-n\_1-n\_2\}\{n\_3\}$ ways.

Multiplying the number of choices at each step results in:

:$\{N\; \backslash choose\; n\_1\}\{N-n\_1\backslash choose\; n\_2\}\{N-n\_1-n\_2\backslash choose\; n\_3\}\backslash cdots=\backslash frac\{N!\}\{(N-n\_1)!n\_1!\}\; \backslash cdot\; \backslash frac\{(N-n\_1)!\}\{(N-n\_1-n\_2)!n\_2!\}\; \backslash cdot\; \backslash frac\{(N-n\_1-n\_2)!\}\{(N-n\_1-n\_2-n\_3)!n\_3!\}\backslash cdots.$

Cancellation results in the formula given above.

File:Multinomial theorem mississippi.svg

The multinomial coefficient

:$\backslash binom\{n\}\{k\_1,\; \backslash ldots,\; k\_m\}$

is also the number of distinct ways to permute a multiset of {{mvar|n}} elements, where {{mvar|k{{sub|i}}}} is the multiplicity of each of the {{mvar|i}}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

:$\{11\; \backslash choose\; 1,\; 4,\; 4,\; 2\}\; =\; \backslash frac\{11!\}\{1!\backslash ,\; 4!\backslash ,\; 4!\backslash ,\; 2!\}\; =\; 34650.$

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.

• Multinomial distribution
• Stars and bars (combinatorics)

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