Question

There is a five-volume dictionary among $50$ books arranged on a shelf in random order. If the volumes are not necessarily kept side by side, the probability that they occur in increasing order from left to right is

• A. $\frac{1}{5!}$
• B. $\frac{1}{5^{50}}$
• C. $\frac{1}{{50}^5}$
• D. None of these

Answer 1

The correct option is A

$\frac{1}{5!}$

Explanation for the correct answer:

Compute the probability:

The total number of ways of arranging $50$ books on the shelf $=50!$

Out of the $50$ places , $5$ places can be chosen for the five volume dictionary in ${}^{50}{C}_5$ ways.

The remaining $45$ books can be arranged in $45$ places in $45!$ ways.

$\therefore$ Number of favorable outcomes $={}^{50}{C}_5\times 45!$

$\therefore$ Required probability$=\frac{{}^{50}{C}_5\times 45!}{50!}$

$=\frac{{\displaystyle \frac{50!}{45!5!}}\times 45!}{50!}$

$=\frac{1}{5!}$

Hence, the required probability is $\frac{1}{5!}$.

Hence, option $\left(A\right)$is the correct answer.