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I came across a probability question from the interview book and does not have a solution from the top of my head. I was thinking of directly counting the number of choices for each user, but it will incur a lot of cases to discuss in separate whether the user's potential mutual best friend has already been taken or not. Can anyone give me some suggestions on how to approach this question? The question is pasted below:

There are two groups of n users, A and B, and each user in A is friends with those in B and vice versa. Each user in A will randomly choose a user in B as their best friend and each user in B will randomly choose a user in A as their best friend. If two people have chosen each other, they are mutual best friends. What is the probability that there will be no mutual best friendships?

Thanks!

Caprikuarius
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  • @lulu Can't multiple people in $A$ choose the same best friend? – angryavian Dec 20 '20 at 00:13
  • @angryavian Ah, maybe I have not understood the question. Thanks. I'll delete my prior comment. – lulu Dec 20 '20 at 00:15
  • I have linked this to a duplicate. Note that the "choice of best friends" defines functions $f:A\to B$ and $g:B\to A$ so your problem is asking for fixed points of $f(g(n))$ as in the linked question. – lulu Dec 20 '20 at 00:25

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