Question

If n letters are placed into n addressed envelopes at random, the probability that atleast one letter will go into wrongly addressed envelope is

• A. $\frac{1}{n}$
• B. $\frac{n-1}{n}$
• C. $1-\frac{1}{n!}$
• D. $\frac{1}{n!}$

Answer 1

Answer: (C)

$1-\frac{1}{n!}$

The total number of permutations we have is $n!$
The number of ways in which all letters are placed in the correct envelope is $1$
For atleast 1 letter to be placed in the wrong envelope, we want the compliment of this set
Hence, $n!-1$
Hence, the probability is $\frac{n!-1}{n!}$
$=1-\frac{1}{n!}$