# Compute $\sum_{i_n=1}^m\sum_{i_{n-1}=1}^{i_n}\cdots\sum_{i_1=1}^{i_2}\sum_{i_0=1}^{i_1}1$

Compute $$\sum_{i_n=1}^m\sum_{i_{n-1}=1}^{i_n}\cdots\sum_{i_1=1}^{i_2}\sum_{i_0=1}^{i_1}1$$

I found this problem in a book that I've been trying go through labeled as "hard" I'm stuck trying to start the problem, let alone even compute it.

Thanks

The sum is equivalent to the number of tuples $(i_0,i_1,\dotsc,i_n)$ where $$1\leq i_0\leq i_1\leq\dotsb\leq i_{n-1}\leq i_n\leq m.$$ This number in turn is equivalent to the number of multisets of cardinality $n+1$ chosen from a set of size $m$ i.e. $$\left(\!\!{m\choose n+1}\!\!\right)=\binom{m+n}{n+1}$$ by stars and bars.