So here's the problem:
- You have a room with n people
- What's the probability that at least one pair in the room will have birthdays that are exactly one day apart from each other?
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As stated, the problem probably needs the inclusion-exclusion principle. However, if you change it to "at most one day apart", then it suffices to find the number of ways to sit $n$ people on $365$ chairs in a line such that no two people are in adjacent chairs, and then divide it by the number of ways to put $n$ people on those chairs without restriction.
If you really want "exactly one day apart", then you can divide into cases, where case $k$ is that the $n$ people have exactly $k$ distinct birthdays (possibly shared). The number of ways to divide $n$ people into $k$ non-empty groups is a Stirling number of the second kind. Each way corresponds to $k!$ ways of sitting $n$ people on exactly $k$ labelled chairs, and we can use the solution to the problem in the above paragraph to find how many ways we can choose those $k$ chairs from the original $365$ chairs.
By the way, in real life the probability you're asking for is either $0$ or $1$, especially since you've already asked all of them for their birthdays, and hence can determine with absolute certainty whether or not two of them have birthdays exactly one day apart. =)