164120

rectangles can have varies of length, the type's of length is from the smallest size to the biggest size. the smallest size is always 1 and the biggest size is any number so i'll call it N. in the example N=10. there are going to be N rectangles for size 1, for size 2 there is going to be N-1 because when you look at the right side size 2 rectangle can fit in the counting stick all the way up to the 2nd last rectangle but if you go further then it won't fit. i'm now going to make a number that can represent any size of rectangle and i'll call it K. K can be 1 or N or anything in between. to find the amount of K sized rectangles that can fit in the N sized stick, we look at the left-most position of the rectangle and there are N-(K-1) such positions. so to find the total number of rectangles you do N+(N-1)+(N-2)+ etc. +1. for N=10 it's 10+9+8+7+6+5+4+3+2+1=55.