Question

A five-digit number divisible by $3$ is to be formed using the digit $0,1,2,3,4,5$ without repetition, The number of ways this can be done is _______

• A. $216$
• B. $184$
• C. $256$
• D. $225$

Answer 1

The correct option is A

$216$

Finding the number of ways for different conditions:

According to the question,

There are two possibilities.

Case1: When digits are $0,1,2,4,5$.

Zero cannot be filled in the first place

$\therefore$Number of ways to form a five-digit number $\begin{array}{rcl}& =& 4\times 4\times 3\times 2\times 1\end{array}$

$=96ways$ _________$\left(1\right)$

Case2: When digits are$1,2,3,4,5$

Number of ways to form a five-digit number $=5\times 4\times 3\times 2\times 1$

$=120ways$ _________$\left(2\right)$

Now add $\left(1\right)\&\left(2\right)$ we get,

Total number of ways$=96+120$

$=216$

Hence, Option (A) is the correct answer.