A five digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without repetitions.
Question:

A five digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without repetitions. The total number of ways this can be done is

(a) 216

(b) 600

(c) 240

(d) 3125

Solution:

Five digit number is to be formed from 0,1,2,3,4 and 5.

Such that number is divisible by 3.

Any number is divisible by 3 if sum of its digits out of these digits,

1, 2, 3, 4, 5 and 0, 1, 2, 4, 5 sum up to be a multiple of 3 

→ for 1,2,3,4,5,

The number of ways a five digit number which is divisible by 3 is 5 × 4 × 3 × 2 × 1                (∵ no restriction is there)

→ for 0,1, 2, 4, 5

The number of ways a five digit number which is divisible by 3 is 4 × 4 × 3 × 2 × 1 = 96     (∵ 0 cannot be placed or first place) 

∴ Total number formed

= 120 + 96

= 216

Hence, the correct answer is option A.

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