Proving sums of multinomial coefficients If m and n are positive integers, how do I prove:
$$\sum_{k_1+\ldots+k_m=n}\binom{n}{k_1,\ldots,k_m}=m^n\;.$$

This is the argument by Jack D'Aurizio spelled out. Use the multinomial theorem
$$
(x_{1} + \dots + x_{m})^{n}
=
\sum_{k_1 + \dots + k_{m} = n} \binom{n}{k_1, k_2, \dots, k_{m}} x_{1}^{k_{1}} \cdots x_{m}^{k_{m}}
$$
and set all $x_{1} = 1$.
