Question

A five digit number is formed by writing the digits $1,2,3,4,5$ in a random order without repetitions. Then, the probability that the number is divisible by $4$ is

• A. $\frac{3}{5}$
• B. $\frac{18}{5}$
• C. $\frac{1}{5}$
• D. $\frac{6}{5}$

Answer 1

The correct option is C

$\frac{1}{5}$

Explanation for the correct option:

Step1. Find the total number of outcomes

Without repetition form a five-digit number using digits $1,2,3,4,5$

Total number of outcomes $=5!$

Step2. Find the number of favorable outcomes condition

Condition for a number to be divisible by $4$

The last two digits should be divisible by $4$

According to the question,

If the last two digits are $12,24,32,52$

Number of ways $3$Places filled $=3!$

Total number of favorable outcomes $=3!\times 4$

Step3. Calculate required probability

$$\begin{gathered} \mathrm{Required}\mathrm{Probability}=\frac{3!\times 4}{5!}\left[\mathbf{\because }\mathbf{Required}\mathbf{}\mathbf{Probability}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{Favourable}\mathbf{}\mathbf{outcomes}}{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{outcomes}}\right] \\ =\frac{1}{5} \end{gathered}$$

Hence, Option(C) is the correct answer.