Suppose you have 10 shirts, all different. When you take 2 shirts out of your closet, you are leaving 8 shirts behind. If you know the 8 shirts left behind, you have uniquely identified the 2 you took out; and if you know the 2 you took out, you know exactly which set of 8 you left behind. So there are exactly as many ways to take out 2 shirts as there are ways to to leave 8 behind.

Put more abstractly, there is a one-to-one correspondence between subsets of size 2 taken from a set of size 10 and subsets of size 8 taken from the same set.