A Universe of Sorts

$nC2$ 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].

§ Intuition for Why Choosing Closed-closed Intervals of [1..n] Is (n+1)C2(n+1)C2(n+1)C2