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I was at dinner and we started discussing this problem:

4 people sitting in chairs at dinner, and various people leave to go to the bathroom.

How many total unique permutations are there of people sitting at the table if a unique permutation also depends on which chair you are sitting in (so a unique permutation depends on who is there, and which chair they are sitting in)

I'm not sure whether it would just be 2^4 (the number of combinations of people, including/excluding each of them) times 4! (the rotations in chairs) but then we were trying to figure out if this is over counting or under counting.

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The formula is given by : \begin{gather} \sum_{i=1}^4 {{4}\choose{i}}^2 i! = 208 \end{gather}

Explantations: For every possible number i of people around the table, there is ${{4}\choose{i}}$ ways to choose which people are around the table, ${{4}\choose{i}}$ to choose the used chairs and $i!$ ways to permute the people with the chairs.

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  • $\begingroup$ Nice answer. However, the boldface word should be Explanation. $\endgroup$
    – N. F. Taussig
    Jul 31, 2021 at 9:50

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