Inequality with two binomial coefficients

Question

I am having trouble seeing why

$$ \binom{k}{2} + \binom{n - k}{2} \le \binom{1}{2} + \binom{n - 1}{2} = \binom{n - 1}{2} $$

Answer 1

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 $k=n$.

Answer 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 $0<k<n$.