I'm trying to derive the binomial distribution function in terms of balls in an urn.
Say the urn has a total number of $N$ balls, where $K$ of them are green and $N-K$ of them are red. Thus the probability of selecting a green ball is $\frac{K}{N}$ and the probability of selecting a red ball is $\frac{N-K}{N}$. Now, say I select $n$ balls with replacement. The number of green balls in my selection will follow a binomial distribution, with probability mass function $$ p(k; n) = \binom{n}{k} \Big(\frac{K}{N}\Big)^k \Big(\frac{N-K}{N}\Big)^{n-k}. $$ Expanding the binomial coefficient and multiplying the factors in the denominator, this can be rewritten as $$ p(k; n) = \frac{K^k (N-K)^{n-k} n!}{N^n k! (n-k)!}. $$ Now, my question is: how should this be interpreted combinatorially? I sort of understand the numerator as selecting $k$ green balls and $n-k$ red balls with replacement, then ordering the whole selection, yielding the number of favorable outcomes. But what about the denominator?
Can anyone give me some insight on this?