Editing Binomial coefficient (section)

=== Generalization to multinomials ===
{{main|Multinomial theorem}}
Binomial coefficients can be generalized to '''multinomial coefficients''' defined to be the number:
: <math>{n\choose k_1,k_2,\ldots,k_r} =\frac{n!}{k_1!k_2!\cdots k_r!}</math>
where
: <math>\sum_{i=1}^rk_i=n.</math>

While the binomial coefficients represent the coefficients of {{math|(''x'' + ''y'')<sup>''n''</sup>}}, the multinomial coefficients
represent the coefficients of the polynomial
: <math>(x_1 + x_2 + \cdots + x_r)^n.</math>
The case ''r'' = 2 gives binomial coefficients:
: <math>{n\choose k_1,k_2}={n\choose k_1, n-k_1}={n\choose k_1}= {n\choose k_2}.</math>

The combinatorial interpretation of multinomial coefficients is distribution of ''n'' distinguishable elements over ''r'' (distinguishable) containers, each containing exactly ''k<sub>i</sub>'' elements, where ''i'' is the index of the container.

Multinomial coefficients have many properties similar to those of binomial coefficients, for example the recurrence relation:
: <math>{n\choose k_1,k_2,\ldots,k_r} ={n-1\choose k_1-1,k_2,\ldots,k_r}+{n-1\choose k_1,k_2-1,\ldots,k_r}+\ldots+{n-1\choose k_1,k_2,\ldots,k_r-1}</math>
and symmetry:
: <math>{n\choose k_1,k_2,\ldots,k_r} ={n\choose k_{\sigma_1},k_{\sigma_2},\ldots,k_{\sigma_r}}</math>
where <math>(\sigma_i)</math> is a [[permutation]] of (1, 2, ..., ''r'').