Problem Solving

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

A. 15
B. 96
C. 120
D. 181
E. 216

[spoiler]OA:E[/spoiler]

Anaira Mitch wrote:How many five- digit numbers can be formed using digits 0,1,2,3,4,5, Which are divisible by 3, without any of the digits repeating?

A. 15
B. 96
C. 120
D. 181
E. 216

If the sum of the digits of integer N is a multiple of 3, then N itself is a multiple of 3.

Adding 5 of the digits above, there are 2 ways to get a sum that is a multiple of 3 if no digit is repeated:
1+2+3+4+5 = 15 and 0+1+2+4+5 = 12.

Number of ways to arrange 1,2,3,4,5 = 5! = 120.

Number of 5-digit integers composed of 0,1,2,4,5:
Ten-thousands digit can be 1,2,4,5 = 4 choices.
Number of ways to arrange the remaining 4 digits = 4! = 24.
Combining our choices for the digits, we get:
Number of possible integers = 4*24 = 96.

Thus, total possible integers = 120+96 = 216.

The correct answer is E.