Question

$5$ letters are to be placed in $5$ addressed envelopes at random. The probability that exactly two letters will go wrong is

• A. $\frac{1}{12}$
• B. $\frac{55}{120}$
• C. $\frac{65}{120}$
• D. $\frac{45}{120}$

Answer 1

The correct option is A $\frac{1}{12}$

Total 5 letters which are to be placed in 5 addresses envelopes. The total number of ways of doing it is equal to $5\ast 4\ast 3\ast 2\ast 1=5!=120$.
Now, to find the favorable number of cases, we have to find the number of ways of placing exactly two letters wrongly. Out of 5 letters, any 2 letters can be chosen to be placed wrongly, which can be done in ${}^5{C}_2=10$ number of ways.
Hence probability is equal to $10/120=1/12$.