Editing Multinomial theorem

{{short description|Generalization of the binomial theorem to other polynomials}}

{{More citations needed|date=December 2022}}
In [[mathematics]], the '''multinomial theorem''' describes how to expand a [[power (mathematics)|power]] of a [[Summation|sum]] in terms of powers of the terms in that sum. It is the [[generalization]] of the [[binomial theorem]] from [[Binomial (polynomial)|binomial]]s to [[Polynomial|multinomial]]s.

==Theorem==
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:
<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'''.<ref>{{citation |first=Martin |last=Aigner |author-link=Martin Aigner |title=Combinatorial Theory |year=1997 |publisher=Springer |page=77 }}</ref> 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}}.<ref name="EC1">{{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}}</ref>{{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|even when {{mvar|x}} equals zero]].}}

In the case {{math|1=''m'' = 2}}, this statement reduces to that of the [[binomial theorem]].<ref name="EC1" />

===Example===
The third power of the trinomial {{math|''a'' + ''b'' + ''c''}} is given by
<math display="block">
(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.
</math>
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 <math>a^2 b^0 c^1 </math> has coefficient <math>{3 \choose 2, 0, 1} = \frac{3!}{2!\cdot 0!\cdot 1!} = \frac{6}{2 \cdot 1 \cdot 1} = 3</math>, the term <math>a^1 b^1 c^1</math> has coefficient <math>{3 \choose 1, 1, 1} = \frac{3!}{1!\cdot 1!\cdot 1!} = \frac{6}{1 \cdot 1 \cdot 1} = 6</math>, and so on.

===Alternate expression===
The statement of the theorem can be written concisely using [[multiindices]]:

:<math>(x_1+\cdots+x_m)^n = \sum_{|\alpha|=n}{n \choose \alpha}x^\alpha</math>

where

:<math>
\alpha=(\alpha_1,\alpha_2,\dots,\alpha_m)
</math>
and
:<math>
x^\alpha=x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_m^{\alpha_m}
</math>

===Proof===
This proof of the multinomial theorem uses the [[binomial theorem]] and [[Mathematical induction|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

: <math>
\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}
</math>

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

:<math> = \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}}\sum_{k_m+k_{m+1}=K}{K\choose k_m,k_{m+1}}x_m^{k_m}x_{m+1}^{k_{m+1}}</math>
:<math> = \sum_{k_1+k_2+\cdots+k_{m-1}+k_m+k_{m+1}=n}{n\choose k_1,k_2,\ldots,k_{m-1},k_m,k_{m+1}} x_1^{k_1}x_2^{k_2}\cdots x_{m-1}^{k_{m-1}}x_m^{k_m}x_{m+1}^{k_{m+1}}
</math>

which completes the induction.  The last step follows because

:<math>{n\choose k_1,k_2,\ldots,k_{m-1},K}{K\choose k_m,k_{m+1}} = {n\choose k_1,k_2,\ldots,k_{m-1},k_m,k_{m+1}},</math>
as can easily be seen by writing the three coefficients using factorials as follows:

:<math> \frac{n!}{k_1! k_2! \cdots k_{m-1}!K!} \frac{K!}{k_m! k_{m+1}!}=\frac{n!}{k_1! k_2! \cdots k_{m+1}!}.</math>

==Multinomial coefficients==
The numbers
:<math> {n \choose k_1, k_2, \ldots, k_m}</math>
appearing in the theorem are the [[Binomial coefficient#Generalization to multinomials|multinomial coefficients]].  They can be expressed in numerous ways, including as a product of [[binomial coefficient]]s or of [[factorial]]s:
:<math>
{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}
 </math>

===Sum of all multinomial coefficients===
The substitution of {{math|1=''x{{sub|i}}'' = 1}} for all {{mvar|i}} into the multinomial theorem
:<math>\sum_{k_1+k_2+\cdots+k_m=n} {n \choose k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}
= (x_1 + x_2  + \cdots + x_m)^n</math>
gives immediately that 
:<math>
\sum_{k_1+k_2+\cdots+k_m=n} {n \choose k_1, k_2, \ldots, k_m} = m^n.
</math>

===Number of multinomial coefficients===

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}}''}}:
:<math>
\#_{n,m} = {n+m-1 \choose m-1}.
</math>

The count can be performed easily using the method of [[Stars and bars (combinatorics)|stars and bars]].

===Valuation of multinomial coefficients===
The largest power of a prime {{mvar|p}} that divides a multinomial coefficient may be computed using a generalization of [[Kummer's theorem]].

=== Asymptotics ===
By [[Stirling's approximation]], or equivalently the [[Gamma function|log-gamma function]]'s asymptotic expansion, <math display="block">\log\binom{kn}{n, n, \cdots, n} = k n \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^3n^3} - \frac{k^6 - 1}{1260k^5n^5} + O\left(\frac{1}{n^6}\right)</math>so for example,<math display="block">\binom{2n}{n} \sim \frac{2^{2n}}{\sqrt{n\pi }}</math>

==Interpretations==

===Ways to put objects into bins===
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.<ref>{{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}}</ref>

===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  {{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 <math>\tbinom{N}{n_1}</math> ways.
*From the remaining {{math|''N'' − ''n''{{sub|1}}}} items choose {{math|''n''{{sub|2}}}} to label 2.  This can be done <math>\tbinom{N-n_1}{n_2}</math> 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 <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===
[[File:Multinomial theorem mississippi.svg|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 [[permutation|permute]] a [[multiset]] of {{mvar|n}} elements, where {{mvar|k{{sub|i}}}} is the [[Multiplicity (mathematics)|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
:<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.

==See also==
* [[Multinomial distribution]]
* [[Stars and bars (combinatorics)]]

==References==
{{notelist}}
{{Reflist}}

[[Category:Combinatorics]]
[[Category:Factorial and binomial topics]]
[[Category:Articles containing proofs]]
[[Category:Theorems about polynomials]]