Question

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

• A. $\frac{5}{24}$
• B. $\frac{11}{24}$
• C. $\frac{11}{30}$
• D. $\frac{5}{30}$

Answer 1

Answer: (C)

$\frac{11}{30}$

Total number of ways this can happen=$5!(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!})$ by derangement rule
Total possible outcomes =5!
Hence probability =$\frac{5!(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!})}{5!}$
=$(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!})=\frac{44}{120}=\frac{11}{30}$