Question

what is the probablity that exactly k letters are placed correctly out of n letters and n envelopes

Answer 1

The number of ways $k$ or more letters out of $n$ letters and $n$ envelopes are placed on the right envelope is

$N_k= \frac{n!}{(n-k)!}$

The number of ways exactly $k$ letters out of $n$ letters and $n$ envelopes are placed on the right envelope is

$N_{k+1}-N_{k-1}= \frac{n!}{(n-k-1)!} -\frac{n!}{(n-k+1)!} \\ N_{k+1}-N_{k-1}= \frac{n!}{(n-k-1)!}[1 -\frac{1}{(n-k)(n-k+1)} ]\\ N_{k+1}-N_{k-1}= \frac{n!}{(n-k-1)!}[1 -\frac{1}{(n-k)(n-k+1)} ]$

The probability that exactly $k$ letters out of $n$ letters and $n$ envelopes are placed on the right envelope is

$\frac{n!}{(n-k-1)!}[1 -\frac{1}{(n-k)(n-k+1)} ]/n!\\ = \frac{1}{(n-k-1)!}[1 -\frac{1}{(n-k)(n-k+1)} ],k=2,3,...,n-1$