Here's the setup to the problem. Assume there are $5$ types of coupons, and you keep collecting coupons until you meet your condition. What is the probability that the number of trials needed to collect all $5$ types of coupons is done in $x$ trials? In general, is there a closed form for finding the probability of collecting $y_1$ of the 1st type, $y_2$ of the 2nd type, $y_3$ of the 3rd type, $y_4$ of the 4th type, $y_5$ of the 5th type # of coupons in $x$ trials?
What about $C$ types of coupons? $(y_1,y_2,...,y_C)$
Can there be a closed form?