25. Three integers are selected from the set $S=1,2,3, \ldots, 19,20$. Find the number of selections where the sum of the three integers is divisible by 3 .
25. Answer: 384
Partition $S$ into three subsets according to their residues modulo 3: $S_{0}=3,6, \ldots, 18$, $S_{1}=1,4, \ldots, 19$ and $S_{2}=2,5, \ldots, 20$. In order for the sum of three integers to be divisible by 3 , either all three must belong to exactly one $S_{i}$ or all three must belong to different $S_{i}$.
Hence total number of such choices is $\binom{6}{3}+2\binom{7}{3}+6 \times 7 \times 7=384$.