llustration of the equality $\binom{m}{n} = \binom{m}{m-n}$

Question

By definition it is easy to see that:

$$\binom{m}{n} = \binom{m}{m-n}$$

But it seems difficult to me explaining this property with a non-technical example. Does anyone know the name of this property? Any real example to illustrate this equality?

Answer 1

Imagine you have a set of $m$ balls from which you want to pick $n$ balls. The number of ways to do so is, by definition, $\binom mn$. But this is also the same as the number of ways to choose the balls which you do not pick, which is by definition $\binom m{m-n}$.

This kind of technique is very common in combinatorics, and the name for it is called "creating a bijection". In essence, instead of counting what we want to count directly, create a one-to-one correspondence between this thing and something else, and then count that something else. Then we know that by the one-to-one correspondence (also known as a bijection), the size of the two sets that we counted must be the same. The bijection in this case is the one that relates a subset $S$ of the $m$ balls with the set $S'$ of balls not in $S$.