A set of four distinct numbers are chosen from the set {1,2,3…9,10,11}. How many ways can this set of 4 numbers be chosen such that the sum is divisible by 3?

Choosing the numbers {1,2,3,6} is the same as choosing the numbers {6,3,2,1}.

Question

A set of four distinct numbers are chosen from the set {1,2,3…9,10,11}. How many ways can this set of 4 numbers be chosen such that the sum is divisible by 3?

Choosing the numbers {1,2,3,6} is the same as choosing the numbers {6,3,2,1}.

rmpjingwei · Answer

We can split up the set with mods, 1,4,7,10≡1 (mod 3) being one subset, 2,5,8,11≡2 (mod 3) being another, and 3,6,9≡0 (mod 3). Then, we can do the problem with casework.

The combinations of mods that will give a sum divisible by 3 are (1,1,1,0),(1,1,2,2),(1,2,0,0), and (2,2,2,0). Notice that (0,0,0,0) is not included because there are only three numbers divisible by 3 in the original set.

By using casework, there are 12 (4C3×3C1) cases for the (1,1,1,3) and (2,2,2,3) subsets, 36 (4C2×4C2) cases for the (1,1,2,2) subset, and 48 (4C1×4C1×4C2) cases for the (1,2,3,3) subset. Thus, there are 12+12+36+48=108 ways.

anonymous · Answer

why are you posting and answering your own questions??