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

If $0\lt k\lt n$, induction gives us
$$\begin{align}
{k\choose2}+{n+1-k\choose2}&={k\choose2}+{n-k\choose2}+{n-k\choose1}\\
&\le{n-1\choose2}+{n-k\choose1}\\
&\le{n-1\choose2}+{n-1\choose1}\\
&={n\choose2}
\end{align}$$
while ${k\choose2}+{n+1-k\choose2}\le{n\choose2}$ holds trivially if $k=n$.
