Question

$5$ letters are to be placed in $5$ addressed envelopes at random. The probability that all the letters will not go into wrongly addressed envelopes is

• A. $\frac{1}{120}$
• B. $\frac{44}{120}$
• C. $\frac{76}{120}$
• D. $\frac{199}{120}$

Answer 1

Answer: (C)

$\frac{76}{120}$

The number of Derangement of $n$ objects is given by $\left[\frac{n!}{e}\right]$ where $[\cdot ]$ represents the integral part and $e$ is the Euler's Number.
Hence, for $n=5$, number of derangement is $44$
Hence, the probability that there is atleast one letter which is correctly inserted is $1-\frac{44}{120}$
$=\frac{76}{120}$