Yes, and the eighth row from the bottom is the first one that has two arrangements with "wall-to-floor" symmetry.
Actually, there is a formula for the number of ways to build this staircase, but first you need a definition i had never heard of.
The hook length of a block in the staircase is 1 + the number of blocks directly above it + the number of blocks directly to the right. E.g., the hook length of the second from bottom block against the wall is 1+2+2=5.
The formula for the number of ways to build the staircase is
(number of blocks)! / Product of all the hook lengths
=10! / (7 * 5^2 * 3^3 * 1^4)
This formula shows how quickly the number grows as the stairway gets taller: 16 for 3 steps, 768 for 4, and 292,864 for 5
Confirming that last one with a diagram is left as an exercise for the reader ;-)