Mathematics - Permutation & Combination Question with Solution | TestHub

Since only one letters can not be in wrong envelope so that at least two letters are in wrong envelope means all the letters are not in right envelopes.

$\therefore x=6!-1=720-1=719$

$Y=$ number of ways, so that all the letters are in wrong envelops

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

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

$\therefore x-y=719-265=454$