Enumeration of integers are in increasing order which have gaps

Question

I want to solve the following:

Calculate the number of ways of selecting five distinct integers $x_1,x_2,x_3,x_4,x_5$ where $0\leq x_1 \lt x_2 \lt x_3 \lt x_4 \lt x_5 \leq 20$

I think this may be a problem of inclusion/exclusion, but it also might be a stars and bars problem.

Which type of problem is this, and how do I get it into a form that is more standard for such a problem? Thank you for any ideas.

Answer 1

You can just pick $5$ distinct elements from $\{0,1,2,\cdots, 20\}$, they automatically put themselves in the right order.

So we have ${21 \choose 5}$ ways