21. Let $S=\{1,2,3, \ldots, 64,65\}$. Determine the number of ordered triples $(x, y, z)$ such that $x, y, z \in S, x<z$ and $y<z$.
21. Answer: 89440
The number of such ordered triples $(x, y, z)$ with $x=y$ is
$$
\binom{65}{2}=2080 .
$$

The number of such ordered triples $(x, y, z)$ with $x \neq y$ is
$$
2 \times\binom{ 65}{3}=87360 \text {. }
$$

Hence the answer is $2080+87360=89440$.