combinations and permutations help

sorry for the abundance of math questions lately, this will probably be the last for now.

First read:
A 5 x 6 rectangle is formed using 1 x1 squares, 5 going up, 6 across. How many rectangles of various sizes can be found in this figure?

To solve:
[firstly, squares are a special type of rectangles]
Think of the grid as 7 vertical lines and 6 horizontal lines. To form a rectangle you must choose 2 vertical lines and 2 horizontal lines out of 6 possible. Each choice of 2 vertical and 2 horizontal gives a distinct rectangle.
So there are (7 choose 2) = 21 ways to choose the vertical lines
and (6 choose 2) = 15 ways to choose the horizontal lines
So there are 21*15 = 315 rectangles.

So this is done by the method of choosing, if you are familiar with it.

Now my actual question is, if instead it were a 5 x 5 square and it asked you to find how many squares of various sizes that were in the 5 x 5 square,
would you still be able to use choosing to solve? And if yes, how?

*just in case of you don't know, 6 choose 2, can be expressed as n! / (n-r)! r! , or 6! / 4! x 2! = 15

If a square within the 5x5 square is 5x5, there is only one square possible. If a square is 4x4, there are 4 different possible squares. The general pattern is (6 - length)², because there are 6-length ways to move up and down (6-length unused rows blocks vertically when a length x length square is made), and there are 6-length ways to move horizontally. So, the answer is the sum of the series (6-length)² from length=1 to length=5, which gives an answer of 55.

I don't see any nice way of doing this by choosing.

yeah, your way was the same as the solution the paper had.
I was just wondering if like the first question, there was a fast effective way of using choosing to solve.
Thanks!