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I'm not sure if I follow the author's solution. Here are some queries I have regarding it:

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So I understand the 'treating the matched couples as single entities' idea and that we treat it as arranging $(19-n)!$ entities around a table, and then times this by $2^n$ as there are 2 ways of arranging each husband-wife pair. This allows us to find the probability that precisely $n$ couples sit next to each other.

However, how do we know that this guarantees that we get precisely $n$ couples sitting next to each other?

Take $n = 5$, we get $2^5 \cdot (14)!$.

Considering the second term, we have $15$ entities to arrange around a table ($5$ couples, $10$ people who cannot sit next to their spouse). But how does this formulation guarantee that we get precisely 10 people who do not sit next to their spouse? I could, for example, have all 5 matched couples sitting in the first slots, and I now have 9 choices for the next slot, $8$ choices for the next etc... (according to this formulation). There does not seem to be a restriction on the number of couples sitting next to each other.

Where have I gone wrong in my reasoning?

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  • $\begingroup$ In your example of $n=5$, yes, $2^n (14!)$ is not the number of arrangements where exactly $5$ couples sit together. Instead it is number of arrangements where specified set of $5$ couples sit together regardless of whether other couples sit together or not. We are applying Principle of Inclusion Exclusion. $\endgroup$
    – Math Lover
    Commented Apr 4, 2021 at 19:26
  • $\begingroup$ Aha! Makes sense thanks - of course, the inclusion-exclusion principle cancels everything out in the end. $\endgroup$
    – threelinewhip
    Commented Apr 4, 2021 at 19:30

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