The total number of positive integer solutions of \(21 ≤ a + b + c ≤ 25\) is

There are 1,160 positive integer solutions. For a fixed sum $s$, two dividers placed in the $s-1$ gaps between units yield $\binom{s-1}{2}$ ordered positive triples. Adding the counts for $s=21,22,23,24,25$ gives $190+210+231+253+276=1160$. Starting at $s=22$ omits 190 triples and gives 970.