G285 2021 Summer Problem Set Problem 3

Problem

$60$ groups of molecules are gathered in a lab. The scientists in the lab randomly assign the $60$ molecules into $5$ groups of $12$ . Within these groups, there will be $5$ distinguishable labels (Strong acid, weak acid, strong base, weak base, nonelectrolyte), and each molecule will randomly be assigned a label such that teams can be empty, and each label is unique in the group. Find the number of ways that the molecules can be arranged by the scientists.

$\textbf{(A)}\ 5^{60} \qquad\textbf{(B)}\ \frac{60!\cdot 5^{60}}{(12!)^5} \qquad\textbf{(C)}\ \frac{60!\cdot 5^{30}}{(12!)^5} \qquad\textbf{(D)}\ \frac{40!\cdot 5^{60}}{11!(12!)^4} \qquad\textbf{(E)}\ 60!5^{60}$

Solution

We have the number of ways to arrange the $5$ groups is $\binom{60}{12} \binom{48}{12} \binom{36}{12} \binom{24}{12} \binom{12}{12} = \frac{60!}{(12!)^5}$ . Now, since the groups are distinguishable, the subsets make $5^{60}$ ways for the arrangements. The answer is $\boxed{\textbf{(B)}}$

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G285 2021 Summer Problem Set Problem 3

Problem

Solution