# How many rectangle(s) with two “#” is/are there in the figure below?

How many rectangle(s) with two “#” is/are there in the figure below?

I tried to count it and my answer is 36.

My teacher shared the answer and ask us to find the reason why.

He said it is just

$$2 * 3 * 2 * 3$$

I found out that my answer is correct. But, what do these factors mean?

Give numbers on the horizontal (up to down) and vertical (left to right) lines(not squares!).

Then the left/right side of the rectangle must be column $$1,2,3$$/column $$4,5,6$$. So, there are $$9$$ possibilities.

The upper/lower side of the rectangle must be row $$1,2$$/row $$4,5$$. So, there are $$4$$ possibilities.

Then, the whole possibilities are $$3$$6.

The two 2 means there are $$2$$ cases each to determine the upper/lower sides, and two 3 means there are $$3$$ cases each, to determine the left/right sides.

• @PRD I'll say the upper/lower cases(left/right cases are exactly same way). There are 4 possibilities, $(U,L)=(1,4), (1,5), (2,4), (2,5)$. Oct 22 at 17:10