Calculate $a_n = \binom{n}{2} + \binom{2}{n}$

Question

Calculate $a_n = \binom{n}{2} + \binom{2}{n}$

Could you give me a hint how to start solving this equation? How can I expand $\binom{2}{n}$?

Definition of $\binom{a}{b}=\frac{a \cdot (a-1) \cdots (a-b+1)}{b!}$ where $a \in \mathbb{C}$. We don't use Gamma function.

Answer 1

It depends a bit on the context you are coming from, but $\binom{2}{n}$ is $0$ for $n>2$ under the combinatorial definition. I would guess that is the appropriate decision here.

If you've used the gamma function before, it's also possible to use that to generalize the choose function.

Answer 2

Hint: $n\leq 2$ and $n$ has to be poitive integer equal to $0$ or greater that leaves us with $3$ possible options.

Answer 3

$$ a_0=\color{lightgrey}{\binom 02}+\binom 20=0+1=1\\ a_1=\color{lightgrey}{\binom 12}+\binom 21=0+2=2\\ a_2=\binom 22+\binom 22=1+1=2\\ a_3=\binom 32+\color{lightgrey}{\binom 23}=3+0=3\\ a_4=\binom 42+\color{lightgrey}{\binom 24}=6+0=6\\ \vdots$$