# combinatorial argument on why the proof is true

$8.$ ($6$ points) Verify that for any $n\ge k\ge 1$ $$\binom{n}2=\binom{k}2+k(n-k)+\binom{n-k}2\;.$$ $\quad$Then give a combinatorial argument for why this is true.

Hi I need help doing this problem

I know this is a given $\binom{n}{2} = \binom{k}{2} + k \cdot (k-1) \binom{n-l}{2}$

Also what is a combinatorial argument as well

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HINT: Let $X$ be a set of $n$ objects; $\binom{n}2$ is the number of $2$-element subsets of $X$. Now split $X$ into two disjoint subsets $Y$ and $Z$ in such a way that $Y$ contains $k$ objects, and $Z$ contains the other $n-k$ objects. Split the $2$-element subsets of $X$ into those whose elements are both in $Y$, those with one element from $Y$ and one from $Z$, and those whose elements are both from $Z$. How big is each of those three sets of pairs?