How many 4-digit positive integers divisible by 3 can be formed using only the digits {1, 3, 5, 6, 7}, such that no digi...

Question

How many 4-digit positive integers divisible by 3 can be formed using only the digits {1, 3, 5, 6, 7}, such that no digit appears more than once in a number?

A. 24  B. 48  C. 72  D. 12

Accepted Answer

48 four-digit numbers We need to pick 4 distinct digits from {1, 3, 5, 6, 7} and arrange them into a 4-digit number that is divisible by 3. A number is divisible by 3 when its digit-sum is a multiple of 3. We first find which subsets of 4 digits have a digit-sum divisible by 3, then count all 4! = 24 arrangements of each such subset.