Editing Multinomial theorem (section)

==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.