Question

A person writes letters to six friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that all of them are in wrong envelope.

Answer 1

The no.of ways in which all letter be placed in wrong envelopes

$=6!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{-1}{3!}+\frac{1}{4!}-\frac{1}{5!}+\frac{1}{6!})$

$=720(1-\frac{1}{1\times 1}+\frac{1}{2\times 1}-\frac{1}{3\times 2\times 1}+\frac{1}{4\times 3\times 2\times 1}-\frac{1}{5\times 4\times 3\times 2\times 1}+\frac{1}{6\times 5\times 4\times 3\times 2\times 1})$

$=720(1+\frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120}+\frac{1}{720})$

$=360-120+30-6+1$ $=265$