The total number of positive integral solution of $$ 15 < x_{1}+x_{2}+x_{3} <= 20 $$ is equal to
The number of positive integer solutions of \(x_1 + x_2 + x_3 = s\) is \(\binom{s-1}{2}\).
We need \(s = 16, 17, 18, 19, 20\):
\(\binom{15}{2} + \binom{16}{2} + \binom{17}{2} + \binom{18}{2} + \binom{19}{2} = 105 + 120 + 136 + 153 + 171 = 685\)
Answer: (A) 685
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