11. Let $S=\{1,2,3, \ldots \ldots, 30\}$. Determine the number of vectors $(x, y, z, w)$ with $x, y, z, w \in S$ such that $x<w$ and $y<z<w$.
11. Answer: 90335
There are two cases to consider: Case (1) $x \in\{y, z\}$ and Case (2): $x \notin\{y, z\}$. For Case (1), there are $2\binom{30}{3}$ ways and for Case (2), there are $3\binom{30}{4}$ ways. Hence, total number of ways $=90335$.