Inequality with two binomial coefficients I am having trouble seeing why 
$$
\binom{k}{2} + \binom{n - k}{2} \le \binom{1}{2} +   \binom{n - 1}{2} = \binom{n - 1}{2}
$$

Assuming $1\le n$ and $0\le k\le n$ it's equivalent to
\begin{align*}k(k-1)+(n-k)(n-k-1)&\le(n-1)(n-2)\\
(n-k)(n-k-1)&\le(n-k-1+k)(n-2)-k(k-1)\\
(n-k)(n-k-1)&\le(n-k-1)(n-2)+k(n-2)-k(k-1)\\
0&\le(n-k-1)(k-2)+k(n-k-1)\\
0&\le(n-k-1)(k-1)\end{align*}
So it holds for all $0<k<n$.
