30. Find the number of subsets $\{a, b, c\}$ of $\{1,2,3,4, \ldots, 20\}$ such that $a<b-1<c-3$.
30. Answer: 680
For any 3-element subset $\{a, b, c\}$, define a mapping $f$ by
$$
f(\{a, b, c\})=\{a, b-1, c-3\} .
$$

Now observe that $\{a, b, c\}$ is a subset of $\{1,2,3,4, \ldots, 20\}$ with $a<b-1<c-3$ if and only if $f(\{a, b, c\})$ is a 3-element subset of $\{1,2,3, \ldots, 17\}$. Hence the answer is $\binom{17}{3}=680$.