How do I count number of ways of drawing items with certain number of duplicates? How do I count the number of ways of drawing K items (with replacement) from a pool of N items such that I have at least D copies of each of any E different items.
For example if there are 10 distinct coupons, how many ways can I draw 100 random coupons and have at least 3 copies of any 5 of them.
I initially thought of inclusion-exclusion and start with n^k and then discount the number of ways of not having the required duplicates, but couldn't get a formula. 
Another example: If there are 3 different coupons and I want to draw 5 at random such that there are 2 copies of at least 2 of them
Drawing:
11223 is a success
11123 is a failure
11122 is a success
 A: It seems like you want to count the number of functions from the set $\{1,2,3\dots K\}$ to the set $\{1,2,3\dots N\}$ so that the pre-image of every element of $\{1,2,3\dots K\}$ has size at least $D$.
We can solve this problem using the $D$-associated stirling numbers of the second kind.
The $D$-Associated number of the second kind $S_D(N,K)$ gives you the number of partitions the set $\{1,2\dots N\}$ has into $K$ parts of size at least $D$. 
Look at the problem this way. What you want to count is the number of ways to color the integers from $1$ to $N$ with $K$ colors so that there are at least $D$ integers of each color. To do this you can first divide the integers into $K$ classes and then select which color gets each of the $k$ classes in $k!$ ways.
The number of ways to divide the integers into classes is $S_D(N,K)$.
Hence what you want is $S_D(n,k)\times k!$.

Note that in general it is hard to calculate the $D$-associated striling numbers , although they do follow the recurrence $S_D(N+1,K)=KS_D(N,K)+\binom{N}{D-1}S_D(N-D+1,K-1)$
I wrote a similar answer here
