Question

Kavita is sending $5$ rakhis to her $5$ brothers. In hurry she mixes up the rakhis and the envelopes. What is the probability that none of the rakhis reach to intended brother.

• A. $\frac{7}{32}$
• B. $\frac{15}{30}$
• C. $\frac{8}{11}$
• D. $\frac{11}{30}$

Answer 1

The correct option is D $\frac{11}{30}$

The given problem is a problem of dearrangement. The number of ways such that no item goes to intended place is given by

$n!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{n!})$ there $n=5$

$\Rightarrow$ Total dearrangement $=5!(1-\frac{1}{1}+\frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120})$

$=60-20+5-1$

$=44$

$\Rightarrow$ Total possibilities $={}^5{P}_5=5!=120$

$\Rightarrow$ Probability $=\frac{44}{120}$

$\Rightarrow P=\frac{11}{30}$

Hence, the answer is $\frac{11}{30}.$