Number of combinations of size $k$ of elements in a multiset with finite multiplicites

Question

Suppose I have 3 types of balls each in varying amounts, e.g. 3 Red (R) balls, 3 Green (G) balls, and 2 Blue (B) balls. How many combinations of size $k, k\leq n$ where $n$ is the total number of balls (in this case 8), can be made?

For example, suppose $k = 2$, then we can make the combinations: RR, RG, RB, GG, GB, and BB of which there are 6. Note that the set RG and GR are the same in my example.

It seems almost like a stars and bars problem, but I don't think it applies here. Is there a closed form solution to this problem or a way to calculate the number of combinations that doesn't involve exhaustively generating every possible $k$-combination?

My ultimate goal is to sum over all $1\leq k\leq n$, if it's easier to calculate that number rather than for a specific $k$, that would be useful to know too.

Answer 1

You can use generating functions which are not difficult to learn at a rudimentary level.

Generating function for $0-3$ red balls = $$ 1+x +x^2 +x^3$$

Generating function for $0-3$ green balls = $$ 1+x +x^2 +x^3$$

Generating function for $0-2$ blue balls = $$ 1+x +x^2 $$

Next, find the coefficient of $[x^k]$ in the expansion of $$(1+x +x^2 +x^3)^2(1+x+x^2)$$

eg for $k=3$ the answer is $9$

Finally, you can sum up the answers for $k = 1\; to\; n$

Actually, you can get all the information you need by simply expanding the generating function at one go, and extracting whatever coefficients you need