Confusion with $\binom{n}{x}$ = $\binom{n}{y}$

Question

I have been taught that when $\binom{n}{x}$ = $\binom{n}{y}$, either

1. $x$ = $y$ OR

2. $x+y=n$

My question is which one should i be prioritizing first, as both can get me different answers. Apologies if this is too dumb a question for this website.

Answer 1

Consider this.

$$x^2 = 4$$

So what's the value of $x$? Well, $x = \pm 2$.

Does one solution have a priority? No. Both $2$ and $-2$ are equally valid solutions.

That said, the context may place certain restrictions which, in turn, may force us to pick one solution and reject another. Or maybe prefer one over the other.

For example, if $x$ represented the number of people, we know $x$ must be positive. and so, reject the negative value.

It's the same with

$$\binom{n}{x} = \binom{n}{y}$$

Both $x$ and $n - x$ are equally valid solutions for $y$. But then, you might be looking for a value different from $x$. In that case, go with $n - x$.