Question

There are $5$ letters and $5$different envelopes. The number of ways in which all the letters can be put in wrong envelope, is

• A. $119$
• B. $44$
• C. $59$
• D. $40$

Answer 1

The correct option is B

$44$

Explanation for correct option

Total number of letter $5$

Total number of direct envelope $5$

Total number of ways

$$\begin{gathered} =\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}\right]5! \\ =\left[1-1+\frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120}\right]5! \\ =\left[\frac{11}{30}\right]120 \\ =44 \end{gathered}$$

Hence, option (B) is correct