Permutation and Combination - Quant/Math - CAT 2013

Question 4 the day: August 30, 2002
The question for the day is from the topic of permutation and combination.

How many five digit numbers can be formed using the digits 0, 1, 2, 3, 4 and 5 which are divisible by 3, without repeating the digits?

(1)

15

(2)

96

(3)

216

(4)

120

Solution:

There are six digits - 0, 1, 2, 3, 4 and 5. To form 5-digit numbers we need exactly 5 digits. So we should not be using one of the digits.

The sum of the numerals 0, 1, 2, 3, 4 and 5 is 15. We know that a 5-digit number is divisible by 3 if an only if the sum of its digits are divisible by '3'. Therefore, we should not use either '0' or '3' while forming the five digit numbers - only then will it be divisible by '3'.

If we do not use '0', then the remaining 5 digits can be arranged in 5! ways = 120 numbers.

If we do not use '3', then the remaining arrangements that are possible without '0' being the first digit are 5! - 4! = 120 - 24 = 96 numbers.

Therefore, there are a total of 120 + 96 such numbers that exist.