Coefficient of $x^{41}$ in $(x^5 + x^6 + x^7 + x^8 + x^9)^5$ 
What is the coefficient of coefficient of $x^{41}$ in $(x^5 + x^6 + x^7 + x^8 + x^9)^5$?

Using summation of G.P., this is equivalent to finding the coefficient of $x^{41}$ in
$$\left(x^5 \left(\frac{1-x^5}{1-x}\right)\right)^5$$
and thus finding coefficient of $x^{16}$ in $(\frac{1-x^5}{1-x})^5$. How to proceed after this?

In other words, how many ways are there to write
$$ 41 = a + b+c+d+e $$
where $a,b,c,d,e$ are integers between $5$ and $9$, inclusive.
The highest sum we can make is $9\cdot 5=45$, so to get $41$ instead we need to remove $4$ units from among the 5 variables. Fortunately, even removing all $4$ of them from the same variable still leaves $5$, which is a valid value. So what we're counting is the same as

How many ways are there to write $4$ as an (ordered) sum of $5$ non-negative integers?

This is a standard combinatorial problem; by the stars-and-bars formula, the answer is $$\binom{4+5-1}{4} = 70$$
