Count no. Of ways

Question

If $n$ identical balls put into $m$ identical boxes, how many ways it can be done, provided that boxes may be empty and all balls have to be put into these boxes at each time.

Answer 1

If the boxes are distinguishable (box 1, box 2, ..., box m) then this is the same as the number of monomials in $m$ variables of degree $n$, and is equal to $\left( \begin{array}{ccc} m+n-1 &\\ n & \end{array} \right)$

Here is a really nice proof/explanation: http://murphmath.wordpress.com/2012/08/22/counting-monomials/

If the boxes are indistinguisable, then Integer partition with fixed number of summands but without order answers this.