Number of ordered ways to pick 3 balls out of an urn with 2 black and 3 white balls without replacement The answer is 7: WWW, WWB, WBW, BWW, WBB, BWB, BBW. Is there an elegant way of calculating it in a way that scales to picking M balls out of K black and N white balls?
 A: There isn't really a nice expression for it, no. What you want in general is a sum of binomial coefficients
$$
\sum_{i = \max(0, M-N)}^{\min(K, M)}\binom{M}{i} 
$$
where each term is the number of orders in which you can pick $i$ black balls and $M-i$ white balls, and the boundaries of the sum are the given by how many black balls you could possibly draw.
A sum of consecutive binomial coefficients like this has no nice closed form in general. For the three cases $\sum_{i = 0}^M$, $\sum_{i = 0}^{M/2}$ and $\sum_{i = M/2}^M$ (in other words, either there are more than enough balls of both colors, or exactly one of the colors is limited, to exactly $M/2$) there is a nice expression (it's just a power of $2$), and cases close to those can be calculated from there, but that's the only special cases that I know of.
A: $$\sum_{k=\max\{0,M-N\}}^{k=\min\{K,N\}}\dfrac{ M!}{k!(M-k)!}$$
For each $k$ balls of the color with smaller number chosen, $(M-k)$ balls of the other color is chosen. Order the balls, account for the identical objects.
A: The generating function for all words without restrictions is
$GF(x,y,t) = {1 \over 1 - xt - yt}$
where the coefficient of $x^ny^kt^m$ encodes the number of word of length $m$ that contain $n$ letters W and $k$ letters B.
$$\sum_{0 \leqslant n}^{N} coeff(x^n)\ \sum_{0 \leqslant k}^{K} coeff(y^k)\ \sum_{m=M}coeff(t^m) \ GF(x,y,t) $$ 
for the given case, this "viewport" will extract the polynomials:
$x^3+3x^2y+3xy^2+y^3$
$x^3+3x^2+3x+1$
$7$
A: Oh dear me !
here is a nice one :
We have to partition in two subsets a string of length M $x_1x_2...x_M$
then the exponential generating function is :
$$ (1 + x + {x^2 \over 2!} + \cdots + {x^K \over K!}).(1 + x + {x^2 \over 2!} + \cdots + {x^N \over N!}) $$
Just take the coefficient of $x^M \over M!$
see here How many arrangements of k items from n items when there are duplicate items
