We wish to distribute 12 indistinguishable stones among 4 distinguishable boxes $B_{1}$, $B_{2}, B_{3}, B_{4}$. (It is permitted some boxes are empty.)

(a) Over all ways to distribute the stones, what fraction of them have the property that the number of stones in every box is even?

(b) Over all ways to distribute the stones, what fraction of them have the property that the number of stones in every box is odd?
We will invoke the following famous theorem, colloquially called stars and bars: the number of ways to distribute $a$ indistinguishable objects among $b$ boxes is $\left(\begin{array}{c}a+b-1 \\ b-1\end{array}\right)$. One can see this by considering a string of $a$ objects ("stars") and $b-1$ dividers ("bars"), which give a distribution into boxes.

The total number of ways to distribute the stones overall is equal to $\left(\begin{array}{c}12+3 \\ 3\end{array}\right)=455$, by stars-and-bars argument.

For (a), we imagine distributing the stones in pairs; we are distributing 6 doublestones. So the number of ways is $\left(\begin{array}{c}6+3 \\ 3\end{array}\right)=84$. Thus the fraction is $\frac{84}{455}=\frac{12}{65}$.

For (b), we also distribute the stones in pairs, but we need to place one stone in each box first: thus, we are distributing 4 double-stones (with the eight remaining stones). The number of ways to do this is $\left(\begin{array}{c}4+3 \\ 3\end{array}\right)=35$. Thus the fraction is $\frac{35}{455}=\frac{1}{13}$.