The total number of ways of selecting two numbers from the set {1,2,3,4, ...,3n} so that their sum is divisible by 3 is equal to

Given number can be grouped as1,4,7,…,3n−2→3λ−22,5,8,…,3n−1→3λ−13,6,9,…,3n→3λThat means, we must take two numbers from last row or one number each from first and second rows. Therefore, the total number of ways is nC2+nC1×nC1=n(n−1)2+n2=3n2−n2