The number of ways of 5 letters put in 5 addressed envelopes so that one letter is in right envelop and all the remaining are placed in wrong envelopes is

    We can select one letter form 5 letters in 5 ways and this can be placed in the right addressed envelope in 1 way.

   Now the remaining 4 letters can be placed in wrong envelopes in $4!\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}\right]$ ways

            $The number of ways=$     $5.4!\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}\right]=45$