Question

Five digit numbers divisible by $3$ are formed using the digits $0,1,2,3,4,5,6,7$ without repetition. The number of such numbers are?

• A. $936$
• B. $480$
• C. $600$
• D. $216$

Answer 1

The correct option is C

$600$

Finding the number of such numbers:

Given, numbers are $0,1,2,3,4,5,6,7$

Now if the five digit number are divisible by $3$ then their sum also divisible by $3$.

So arranging the digits to from such five digit numbers, we have;

If sum is $21$ then leaving number $[0,2]$, we have

$[1,3,4,6,7]$=$5!=120$

If sum is $18$ then leaving number $[2,3]$, we have

$[0,1,4,6,7]$=$4\times 4!=96$

Similarly,

$[7,1]$ : $[0,2,3,4,6]=96$

$[2,6]$ : $[0,1,3,4,7]=96$

$[7,4]$ : $[0,1,2,3,6]=96$

$[1,4]$ : $[0,2,3,6,7]=96$`

Therefore, total number of such numbers

$$\begin{gathered} =120+96+96+96+96+96 \\ =600 \end{gathered}$$

Hence, correct option is $\mathbf{\left(}\mathit{C}\mathbf{\right)}$.