I came across a variation of the birthday problem asking "in a room of $4$ people what is the probability that at least $3$ of them share the same birthday".
I was unsure of the answer and thought that it would be P($3$ share the same birthday) + P($4$ share the same birthday), which equals: $1\cdot\frac{1}{365^2} + 1\cdot\frac{1}{365^3}$, and this comes out to be around $0.0000075$%.
However my friend said that he thinks to correctly calculate the answer, the probability of $4$th person not having the same birthday should be included in the calculation somewhere.
What would be the correct probability of at least $3$ out of $4$ people sharing the same birthday, and how could you extend the problem to work out the probability of at least "$x$" out of "$y$" people having the same birthday?