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

In how many ways one can choose three distinct numbers from the set $\{1,2,3,\dots \dots,19,20\}$ such that their product is divisible by 4? We partition the set $\{1,2,3,\dots \dots,19,20\}$ into three disjoint sets $S_1=\{4,8,12,16,20\},S_2=\{2,6,10,14,18\},S_3=\{1,3,5,7,9,11,13,15,17,19\}$ Three selected (distinct) numbers will not be divisible by $4$ $iff$ all the $three$ numbers are selected form $S_3$ or $two$ of them are selected from $S_3$ and $one$ of them from $S_1$. Numbers of such numbers are $\binom{10}{3}+\binom{5}{2} \times \binom{5}{1} = 345$ So numbers of selection such that their product is divisible by $4$ is $\binom{20}{3}-345= 795$