Question

Four different objects $1,2,3,4$ are distributed at random in four places marked $1,2,3,4$. What is the probability that none of the objects occupy the place corresponding to their number?

• A. $\frac{17}{24}$
• B. $\frac{3}{8}$
• C. $\frac{1}{2}$
• D. $\frac{5}{8}$

Answer 1

Answer: (B)

$\frac{3}{8}$

A derangement is a permutation of objects that leave no object in its original position.

Number of derangement's of set with n elements is

$D_n=n![1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}.........+{(-1)}^n\frac{1}{n!}]$

The probability of derangement = $\frac{D_n}{n!}$

In this case,n=4

Thus,probability that none of the objects occupy the place corresponding to their number

$=1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}=1-1+\frac{1}{2}-\frac{1}{6}+\frac{1}{24}=\frac{3}{8}$