is there a general combinatorics formula like n choose k but instead we choose k,l,m ect.

Question

Say for example we have a box with $n$ balls of which $r$ are red, $b$ are blue, $g$ are green, $y$ are yellow (so $n=r+b+g+y$)

Now we draw out all of them, one after the other, without ever placing them back.

Could we generalize the $n$ choose $k$ formula to say that all possible combinations are $n! \over r!b!g!y!$ ?

Answer 1

As mentioned in the comments, multinomial coefficients are likely what you are looking for. A real-life example where they are useful is:

The number of ways to draw 4 running backs, 5 quarterbacks, and 6 wide receivers from a roster of 15 potential players. The number of arrangements of 15 objects into three bins of sizes 4, 5, and 6 would be given by:

$\binom{15}{4, 5, 6} = 630,630$ arrangements

Now, it can be shown this is equivalent to $n!\over k_1!, ... , k_m!$. I think it makes sense to build up from the intuition of binomial coefficients, whereby multinomial coefficients of the form $\binom{n}{ k_1,...,k_m}$ are equivalent to the product:

$\binom{k_1}{k_1}$ * $\binom{k_1 + k_2}{k_2}$ * $\binom{k_1+k_2+k_3}{k_3}$ * ... * $\binom{k_1 + ... + k_m}{k_m}.$