# A 5-digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4 and 5 without repetition.

Question:

A 5-digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is

(a) 216

(b) 600

(c) 240

(d) 3125

Solution:

(a) 216

A number is divisible by 3 when the sum of the digits of the number is divisible by 3.

Out of the given 6 digits, there are only two groups consisting of 5 digits whose sum is divisible by 3.

1+2+3+4+5 = 15

0+1+2+4+5 = 12

Using the digits 1, 2, 3, 4 and 5, the 5 digit numbers that can be formed = 5!

Similarly, using the digits $0,1,2,4$ and 5, the number that can be formed $=5 !-4 !\{$ since the first digit cannot be 0$\}$

$\therefore$ Total numbers that are possible $=5 !+5 !-4 !=240-24=216$