Question

$4$ customers at a restaurant order $4$ meals. What is the probability that a waiter delivers these $4$ orders to the $4$ customers so that no customer receives what they ordered?

• A. $\frac{2}{3}$
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{2}$

Answer 1

Answer: (B)

$\frac{1}{3}$

Total number of customers are $4$ and the total number of orders are $4$

The total number of ways these $4$ orders are placed to $4$ customers is $4!=24$

The number of derangements are $4!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!})=8$

The probability is $\frac{8}{24}=\frac{1}{3}$

Therefore the correct option is $B$