What is $\dbinom{n}{0}$ for any positive integer $n$?
I know that $\dbinom{n}{k}$ is the number of ways to choose $k$ objects out of $n$ distinct objects, without regard to order. So, $\dbinom{n}{0}$ is the number of ways to choose $0$ objects out of $n$ distinct objects. That means I don't choose any of the objects, I just leave them all behind. There is only one way to do that, so $\dbinom{n}{0} = 1$ for any positive integer $n$.

