Counting problem (combination). How many ways to create 2 indistinguishable piles from 21 identical elements. You have 21 identical elements, let's say books. How many ways to place the books into 2 indistinguishable piles? (This means that putting 16 books in the first pile and 5 books in the second is the same as 5 books in the first pile and 16 books in the second)

Lets say there are $F$ books in the first pile and $S$ books in the second pile
$$F+S=21$$
$$F=21-S$$
Since $F,S \geq 1$,
for each $S,1\leq S\leq 20 $  there is a  corresponding $F$ pile.
This gives $20$ solutions.
But each case was overcounted twice.
Hence there are a total of $10$ ways
