Question

Five-digit numbers divisible by $3$ are to be formed using the numerals$0,1,2,3,4$ and $5$, without repetition. The total number of ways this can be done is equal to:

• A. $216$
• B. $240$
• C. $600$
• D. $3125$

Answer 1

The correct option is A

$216$

Step 1: Find the total number of possible ways.

We have been given five-digit numbers divisible by $3$ are to be formed using the numerals$0,1,2,3,4$ and $5$, without repetition.

We need to find the total number of ways this can be done.

The number of five-digit numbers except zero would be,

$$\begin{gathered} \Rightarrow 5!=5\times 4\times 3\times 2\times 1 \\ \Rightarrow 5!=120...................\left(i\right) \end{gathered}$$

The number of five-digit numbers except three would be,

$$\begin{gathered} =5!-4! \\ =(5\times 4\times 3\times 2\times 1)-(4\times 3\times 2\times 1) \\ =120-24 \\ =96..................\left(ii\right) \end{gathered}$$

From $\left(i\right)\&\left(ii\right),$

The total number of possible ways would be,

$120+96=\mathbf{216}$

Therefore, option (A) is the correct answer.