Question

A person writes letlers to six friends and addresses the corresponding envelopes. Let $x$ be the number of ways so that at least two of the letters are in wrong envelopes and $y$ be the number of ways so that all the letters are in wrong envelopes. Then $x-y=$

• A. 719
• B. 265
• C. 454
• D. None

Answer 1

Answer: (C)

If all the letters are not in the right envelopes, then at least two letters must be in wrong envelopes.
$\therefore x=6!-1=719$
Now $y=$ number of ways so that all the letters are in wrong envelopes
$=6!\left\{1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}+\frac{1}{6!}\right\}$
[Deragement formula]
$=360-120+30-6+1=265$
$\therefore x-y=454$