Doing this leetcode Fun fact: my first approach to this about a month ago was $O(n!)$ with recursion, lets see if I can do better this time.

Let $d$ be the number of dice rolls, $f$ be the number of faces and $t$ be the target sum.

We want to find the coefficient of the $x^t$ term in

$\left(\sum_{i=1}^f x^i\right)^d = x^d \left(\sum_{i=0}^{f-1} x^i\right)^d = x^d\left(\frac{x^f - 1}{x - 1}\right)^d$

Our problem is now reduced to finding the $t-d =: k$ coefficient in

$\left(\frac{x^f - 1}{x - 1}\right)^d$