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Each of the 20 balls is tossed independently and at random into one of the 5 bins. Let $p$ be the probability that some bin ends up with 3 balls, another with 5 balls, and the other three with 4 balls each. Let $q$ be the probability that every bin ends up with 4 balls. What is $\frac{p}{q}$ ? 
Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability
Solution 1: For simplicity purposes, we assume that the balls and the bins are both distinguishable. Recall that there are $5^{20}$ ways to distribute $20$ balls into $5$ bins. We have $$ p=\frac{5 \cdot 4 \cdot\left(\begin{array}{c} 20 \textbackslash\textbackslash{} 3,5,4,4,4 \end{array}\right)}{5^{20}} \text { and } q=\frac{\left(\begin{array}{c} 20 \textbackslash\textbackslash{} 4,4,4,4,4 \end{array}\right)}{5^{20}} \text {. } $$ Therefore, the answer is $$ \frac{p}{q}=\frac{5 \cdot 4 \cdot\left(\begin{array}{c} 20 \textbackslash\textbackslash{} 3,5,4,4,4 \end{array}\right)}{\left(\begin{array}{c} 20 \textbackslash\textbackslash{} 4,4,4,4,4 \end{array}\right)}=\frac{5 \cdot 4 \cdot \frac{20 !}{3 ! \cdot 5 ! \cdot 4 ! \cdot 4 ! \cdot 4 !}}{\frac{20 !}{4 ! \cdot 4 ! \cdot 4 ! \cdot 4 ! \cdot 4 !}}=\frac{5 \cdot 4 \cdot(4 ! \cdot 4 ! \cdot 4 ! \cdot 4 ! \cdot 4 !)}{3 ! \cdot 5 ! \cdot 4 ! \cdot 4 ! \cdot 4 !}=\frac{5 \cdot 4 \cdot 4}{5}=(\mathbf{E}) 16 . $$  Solution 2:  For simplicity purposes, we assume that the balls and the bins are both distinguishable. Let $q=\frac{x}{a}$, where $a$ is the total number of combinations and $x$ is the number of cases where every bin ends up with 4 balls.  We can take 1 ball from one bin and place it in another bin so that some bin ends up with 3 balls, another with 5 balls, and the other three with 4 balls each. Note that one configuration of $4$-$4$-$4$-$4$-$4$ corresponds to $5 \cdot 4 \cdot 4=80$ configurations of $3$-$5$-$4$-$4$-$4$. On the other hand, one configuration of $3$-$5$-$4$-$4$-$4$ corresponds to 5 configurations of $4$-$4$-$4$-$4$-$4$. Therefore, we have $$ p=\frac{80}{5} \cdot \frac{x}{a}=16 \cdot \frac{x}{a}, $$ from which $\frac{p}{q}=$ (E) 16 . 