Question

A person writes letters to his $5$ friends and addresses the corresponding envelopes. Number of ways in which the letters can be placed in the envelope, so that atleast two of them are in the wrong envelopes is ?

Answer 1

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

$=120(1-\frac{1}{1\times 1}+\frac{1}{1\times 2}+\frac{1}{1\times 2\times 3}+\frac{1}{1\times 2\times 3\times 4}+\frac{1}{1\times 2\times 3\times 4\times 5})$

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

$=120\left(\frac{119}{120}\right)$

$=119$