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. $240$
• C. $600$
• D. $3125$

Answer 1

The correct option is A

$216$

Explanation for the correct option:

Step1. Define the divisibility rule for $3$

When the sum of the digits of a number is a multiple of $3$ then the number is divisible by $3$.

Step2. Calculate the number of ways for all possibilities

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

Zero cannot be placed in the first place to form a five-digit number

$\therefore$Number of ways using $0,1,2,4,5$ $=4\times 4\times 3\times 2\times 1$

$=96$ …(1)

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

Number of ways using $1,2,3,4,5$ $=5\times 4\times 3\times 2\times 1$

$=120$ …(2)

Step3. Calculate the total number of ways.

Add (1) and (2)

Total number of ways $=96+120$

$=\mathbf{216}$

Hence, Option (A) is the correct answer.