We want to find the number of upward-pointing subtriangles (of any size) nested inside the big triangle.

In each subtriangle $T$ of size one, we have written the number of subtriangles that have $T$ as the bottom right corner. Thus, the total number of subtriangles is the sum of all the numbers. We saw before that that sum is $\left(\begin{matrix}N+2\\3\end{matrix}\right)$, where $N$ is the number of rows.