§ Intuition for why choosing closed-closed intervals of [1..n] is $(n+1)C2$

• $nC2$ counts all intervals $\{ [i, j]: i > j \}$.
• To count intervals $[i, i]$, there are $n$ of them, so it's $nC2 + n$ which is $n(n-1)/2 + n$, which is $n(n+1)/2$ or $(n+1)C2$.
• Combinatorially, add a "special point *" to [1..n]. If we pick a pair (i, *) from the $(n+1)C2$, take this to mean that we are picking the interval [i, i].