TITLE: Combinations of cards out of a deck with sets
QUESTION [8 upvotes]: I have a deck of $25$ cards, $10$ of them are identical to each other, and the other $15$ are made of $5$ sets of $3$ identical cards for visualisation:
$10$ black, $3$ green, $3$ red, $3$ yellow, $3$ blue, $3$ pink
If I draw $10$ cards randomly, how many different hands can I draw?

I tried using my combinatorics knowledge but it wasn't enough, the answer itself is less important to me but the actual way is what I'm after here.

REPLY [8 votes]: Use the generating function $$(1+x+x^2+x^3)^5(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10})$$ and look at the co-efficient of the $x^{10}$ term.  This generating function could be written as $$\left(\dfrac{1-x^4}{1-x}\right)^5 \dfrac{1-x^{11}}{1-x}$$
I get $1 + 5 + 15 + 35 + 65 + 101 + 135 + 155 + 155 + 135 + 101 = 903$ distinct  hands though these are not equally like.