Proof $\tbinom{n}{2}=\tbinom{k}{2}+k(n-k)+\tbinom{n-k}{2}$ based on binomial coefficient definition

Question

How can I prove this equation using the binomial coefficient definition?

$\tbinom{n}{2}=\tbinom{k}{2}+k(n-k)+\tbinom{n-k}{2}, 1\le k\le n$

So far I've just written the equation using the definition. Any tips would be appreciated!

Answer 1

Hint: Fix $k$ and label the $n$ elements as $\{1,\ldots,n\}$. There are 3 kinds of 2-element subsets from $\{1,\ldots,n\}$:

• Both elements are from $\{1,\ldots,k\}$
• 1 element is from $\{1,\ldots,k\}$ and the other element is from $\{k+1,\ldots,n\}$
• Both elements are from $\{k+1,\ldots,n\}$

Can you compute: how many subsets there are for each of the 3 kinds above?