Proposition 14

Let $n$ be an integer $> 0$. Then the number $a_n$ (resp. $b_n$) of ordered pairs $(i, j)$ of integers such that $1 \leqslant i \leqslant j \leqslant n$ (resp. $1 \leqslant i < j \leqslant n$) is $\frac{1}{2}n(n+1)$ ((resp. $\frac{1}{2}n(n-1)$)).

For $b_n$ is the number of subsets of 2 elements in $[1, n]$; hence

The value of $a_n$ is deduced from this by noting that the set of ordered pairs $(i, j)$ such that $1 \leqslant i \leqslant j \leqslant n$ is the union of the set of ordered pairs $(i, j)$ such that $1 \leqslant i \leqslant j < n$ and the set of pairs $(i, i)$ where $1 \leqslant i \leqslant n$. Thus $a_n = n + b_n = \frac{1}{2} n(n+1)$.