<s>[INST] N Choose K Confusion $$
  \binom{kn}{2} = k\binom{n}{2} + \binom{k}{2} n^2,n,k ≥ 2
$$
Im not sure how to prove algebraically this. Can someone help me?
 [/INST] Since $\binom{m}{2} = \frac{m(m-1)}{2}$, you are seeking to prove:
$$
  \frac{k n (k n -1)}{2} = k \frac{n(n-1)}{2} + n^2 \frac{k(k-1)}{2} \stackrel{\text{expand}}{=} {\color\red{ k \frac{n^2}{2}}}  - k \frac{n}{2} + n^2 \frac{k^2}{2} {\color\red{- n^2 \frac{k}{2}}} \stackrel{\text{factor}}{=} k n \frac{k n -1}{2} 
$$
</s>