TITLE: $\sum_{k=1}^n \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$
QUESTION [6 upvotes]: (Own) Let $n,m$ be positive integers such that $m>n$. Prove that $$\sum_{k=1}^n \sum_{a_1+a_2 + \cdots +a_k=n} \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$$ where $1 \le a_i \; (1 \le i \le k)$ and $a_1+a_2+ \cdots + a_k=n$. 
In here, $b_i$ is the number of $a_x \; (1 \le x \le k)$ that have the same value. For example, if $n=6$ and $6=3+1+1+1$ then $b_i$ are $1$ (one number $3$) and $3$ (three numbers $1$).

REPLY [2 votes]: Suppose you need to produce n numbers, c_1,..., c_n, each between 1 and m.
You can do it in this way -

Select k, number of distinct integers in the list
Choose k distinct numbers between 1 and m
Select number of times a_i those numbers will appear
Choose a matching of the counts a_i to the k numbers, by observing that we need to match for distinct counts - choose b_i numbers out of the k for each distinct count
Choose positions for all these k numbers

This is the lhs.
Also the number of ways is m^n, which is the rhs.