Editing Multinomial theorem (section)

==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>