We've got a pool containing 5 A-balls, 4 B-balls, 3 C-balls, 2 D-balls and one E-ball. How many ways are there to pull out 5 balls?
I thought of dividing off from the formula: $\frac{15!}{10!}$ but the issue is that we don't know which balls in each sampling will be identical.
Is there a nice combinatorial method to solve this problem?
It is the coefficient of $x^5$ in $(1+x)(1+x+x^2)(1+x+x^2+x^3)(1+x+x^2+x^3+x^4)(1+x+x^2+x^3+x^4+x^5)$, which is $\boxed{71}$ by W|A.
Indeed, you can see that to get $x^5$ we need to choose some power from every parenthesis, and each of them corresponds to one of the jars.