Counting Balls / Elementary Generating Functions

Question

I have a quiz tomorrow in an elementary combinatorics class, and I'm trying to understanding these generating function problems. For some reason, I can't see to figure out how to set these two up. Also, when the problem asks "find the number of ways" I just have to find the generating function and then what terms' coefficient I'm interested is - not the actual coefficient itself.

Problem 1: Find the number of ways to select 10 balls from a bin containing 10 balls each of 5 different colors.

Maybe I'm just stupid, but I was having trouble interpreting this question - what is meant by "10 balls each of 5 different colors" ? Wouldn't that mean I have 50 balls, not 10 in the bin, or does it mean that each of the 10 balls could be any of 5 different colors (and if this is the case, I have no idea where to start). I just don't get what it's asking … (maybe I should have posted this in an English forum).

Problem 2: Now, find the number of ways to selected 10 balls when there are 10 balls of one color, but only 8 balls of the other 4 colors.

Here, again, I am shaky on the interpretation. Are there 8 of each remaining color (i.e. 32 balls with 8 of each of the 4 colors) - or are there are 8 balls along the total of 4 colors - and how would you set that up

All the other problems on the practice sheet were really straightforward, so I feel like there isn't anything tricky here - but I don't see how to start.

Answer 1

Let's use generating functions.

1. 10 balls out of a bin having 10 each of 5 colors: Each color can appear up to 10 times, i.e., it's generating function is $1 + z + \cdots + z^{10} = (1 - z^{11}) / (1 - z)$. We want 10 balls in all out of the above. As there are 10 balls of each color, it is the same as having an infinite number of balls: \begin{align} [z^{10}] \left( \frac{1 - z^{11}}{1 - z} \right)^5 = [z^{10}] (1 - z)^{-5} = (-1)^{10} \binom{-5}{10} = \binom{10 + 5 - 1}{5 - 1} = 1001 \end{align}
2. 10 balls out of 10 of one color, 8 of each of the others: As above, ellipsis indicates terms that can't affect the result: \begin{align} [z^{10}] \frac{1 - z^{11}}{1 - z} \left( \frac{1 - z^9}{1 - z} \right)^4 &= [z^{10}] \frac{(1 - z^9)^4}{(1 - z)^5} \\ &= [z^{10}] (1 - 4 z^9 + \cdots) (1 - z)^{-5} \\ &= [z^{10}](1 - z)^{-5} - 4 [z](1 - z)^{-5} \\ &= (-1)^{10} \binom{-5}{10} - 4 \cdot (-1) \binom{-5}{1} \\ &= 981 \end{align}