Question 99
Determine the number of subsets $S$ of $\{1,2, \ldots, 1000\}$ that satisfy the following conditions:
- $S$ has 19 elements, and
- the sum of the elements in any non-empty subset of $S$ is not divisible by 20.
- $S$ has 19 elements, and
- the sum of the elements in any non-empty subset of $S$ is not divisible by 20.