Probability of "L" shape on chessboard

Hint: We assume that the chessboard is the conventional $8\times 8$, and that we have to see an L when we are sitting in the usual position, so an "upside down" L doesn't count.

There are $\binom{64}{3}$ equally likely ways to choose $3$ squares. Now we count the number of choices that give an L.

The top square of the L can be in any one of the top $7$ rows. How many ways are there to place an L with top square in one of these rows? Your "filling out the L to a square" is a good way to visualize.