Question

In how many ways $6$ letters can be placed in $6$ envelopes such that:

No letter is placed in its corresponding envelope.

• A. $265$
• B. $275$
• C. $255$
• D. None of these

Answer 1

The correct option is A $265$

when no letters are in the correct envelope, this is to find out the number of derangements of the $6$ envelops. The number of derangements is given by $n!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+.....+(-1{)}^2\cdot \frac{1}{n!})$

Here $n=6$.

Hence no. of derangements

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

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

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

$=240+30-5$

$=270-5=265$ ways.