There is a bag which contains 10 red beads, 10 green beads and 12 yellow beads. How many ways can you pick a pile containing 22 beads?

Question

At first glance i thought to combine all 32 beads and just pick 22 from them because there's nothing saying you need a certain amount of a specific colour bead, or should I be looking at this question as each colour of bead is a generating function and then finding the coefficient for $[x^{22}]$

Answer 1

When you have $0$ red beads, green+yellow $=22$, so there is only one way. When you have $1$ red bead, green+yellow $=21$, so you have two ways,..., when you have $10$ red beads, green+yellow $=12$, so you have eleven ways. Sum= $66$.

Answer 2

Although you can case bash with 10 cases of red, which isn't terribly slow, complementary counting makes this way easier. You are trying to take 10 beads to not pick from a pile of 10 red, 10 green, and 12 yellow. This is essentially just a stars and bars problem with 10 stars and 2 bars (try to see if you can figure out what to use as your stars and bars), so the answer is 12C2, or 66 ways. Unless this is a specific exercise for generating functions, please do not use them as even the cases method is quite a bit faster.