## Reverse Birthday Problem (Solution)

Let \(g(n,k)\) be the number of ways you can put n people into k buckets:

\[g(n,k) = k^n\]Let \(f(n,k)\) be the number of ways you can put n people into k buckets, *using all k buckets*.

In words, the number of ways you can put n people into k buckets *using all k buckets*, is the number of ways you can put n people into k buckets, minus the number of ways you can put n people into exactly 1 bucket (times the number of ways you can choose that one bucket out of k buckets), minus the number of ways you can put n people into exactly 2 buckets (times the number of ways you can choose those two buckets out of k buckets), etcâ€¦

Now that we have \(g\) and \(f\), the original problem just becomes:

Find the smallest \(N\), s.t. \(f(N, 365) > g(N, 365) / 2\).

Using a computer, I found the smallest \(N\) to be \(2287\).