What is $\dbinom{n}{n}$ for any positive integer $n$?
I know that $\dbinom{n}{k}$ means the number of ways to choose $k$ objects out of $n$ distinct objects, without regard to order. So, $\dbinom{n}{n}$ means the number of ways to choose $n$ objects out of $n$ distinct objects. But that's just one way, since I have to choose all of them. Therefore, $\dbinom{n}{n} = 1$ for any positive integer $n$. 