I am a newbie and I have spent a few days solving my following problem but I can't. So I would like to ask the help from all members here.
My problem is the following:
I have $N$ people and $M$ ($M$ > $N$) chairs (all integers number). Each person will randomly choose one chair and choose independently with each other. Note that two or more people can choose the same chair.
My question is: I would like to compute the probability $E$ (from $M-N$ -> $M-1$) chairs that are not chosen. For sure, case $E = M - N$ and $E = M - 1$ are trivial but for other cases, I stuck.
Thank you very much for all your answers.
Gau
The number of ways to choose exactly $K=M-E$ chairs is: $$ \binom MK{N \brace K}K!, $$ where ${N \brace K}$ is the Stirling number of the second kind.
Therefore the probability you are looking for is: $$ 1-\frac 1{M^N}\binom MK{N \brace K}K!. $$