Question

A five-digit number divisible by $3$ is to be formed using the digits $0,1,2,3$ and $4$ without repetition of digits. What is the number of ways this can be done?

• A. $96$
• B. $48$
• C. $32$
• D. No number can be formed

Answer 1

The correct option is D No number can be formed

If a five digit number has to be formed using $0,1,2,3,4$ without repetition, all five digits must be used.

Since all five numbers will be used, sum of digits of all 5-digit number formed using $0,1,2,3,4=0+1+2+3+4=10$

For a number to be divisible by $3$, sum of its digits must be divisible by $3$.

Since, the sum of digits of all 5-digit number formed using $0,1,2,3,4$ is $10$, none of these 5-digit numbers are divisible by 3.

So, no 5-digit number divisible by 3 can be formed using $0,1,2,3,4$ without repetition.

So, the answer is option (D)