Indian Statistical Institute B.Math & B.Stat : Combinatorics

Let $S = \{(a_1,a_2,a_3)\} | \quad 0 \leq a_i \leq 9 \quad and \quad a_1+a_2+a_3 \quad is \quad divisible \quad by \quad 3 \}$. Find the number of elements in $S$. We divide the integers $0 \leq a_i \leq 9$ into three groups having the property that each element of the same group leaves the same remainder on being divided by $3$. The groups are $\{0,3,6,9\}, \{1,4,7\} \quad and \quad \{2,5,8\}$. Now a $three-tuple$ $(a_1,a_2,a_3)$ will be divisible by $3$ iff and only if each of the co-ordinate belongs to the same group or each of them belongs to the different groups, giving a total of $4^3+3^3+3^3+ 4 \times 3 \times 3 \times 3! = 334$ possible $three-tuples$ which are divisible by $3$.