G285 2021 Summer Problem Set Problem 4

Problem

$16$ people are attending a hotel conference, $8$ of which are executives, and $8$ of which are speakers. Each person is designated a seat at one of $4$ round tables, each containing $4$ seats. If executives must sit at least one speaker and executive, there are $N$ ways the people can be seated. Find $\left \lfloor \sqrt{N} \right \rfloor$ . Assume seats, people, and table rotations are distinguishable.

$\textbf{(A)}\ 720 \qquad\textbf{(B)}\ 1440 \qquad\textbf{(C)}\ 2520 \qquad\textbf{(D)}\ 5760\qquad\textbf{(E)}\ 6172$

Solution

We divide executives and speakers; there are $\frac{8!}{2!2!2!2!} = 2520$ ways to arrange the executives. Similarly, there are $\frac{8!}{2!2!2!2!} = 2520$ ways to arrange the speakers. Finally, there are $\binom{4}{2} = 6$ ways to arrange the executives and speakers. The answer is $2520 \sqrt{6} \implies \boxed{\textbf{(E)}}$

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G285 2021 Summer Problem Set Problem 4

Problem

Solution