17. Given a set $\mathrm{S}=\{1,2,3, \ldots, 199,200\}$. The subset $\mathrm{A}=\{a, b, c\}$ of $\mathrm{S}$ is said to be "nice" if $a+c=2 b$. How many "nice" subsets does $\mathrm{S}$ have?
(Note: The order of the elements inside the set does not matter. For example, we consider $\{a, b, c\}$ or $\{a, c, b\}$ or $\{c, b, a\}$ to be the same set.)
17. Answer: 9900
Since $a+c$ is even, $a$ and $c$ have the same parity (either both odd or both even). There are 100 odd and 100 even numbers in $\mathrm{S}$. Once $a$ and $c$ are chosen, $b$ is determined. The number of "nice" subsets is $2\binom{100}{2}=100(99)=9900$