The game of Cootie, where players roll dice to collect parts of an insect (cootie), is a variant of the coupon collector's problem.
Instead of collecting a single instance of each coupon, players must collect multiple copies (6 legs, 2 eyes, 1 head, etc.) to win. It turns out you can compute the expected number of rolls to win at Cootie (even with a weighted die) with a finite sum.
In particular, if you have $L$ objects to collect and for each object $\ell<L$ you need $q_\ell$ copies and the probability of getting the object is $p_\ell$, then the expected number of rolls to get all of the needed objects is
$\displaystyle\sum_{\ell\in L} \frac{{p_\ell}^{q_\ell}}{(q_\ell -1)!}\int_0^\infty x^{q_\ell}\exp(-x) \prod_{k\in L-\{\ell\}}(\exp(p_k x)-\exp_{<q_k}(p_k x))dx.$
If you're interested, check out my paper out for the full computation.
