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

enter image description here

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?

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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.

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  • $\begingroup$ Thank you for the answer but I did some updates. $\endgroup$
    – PRD
    Oct 22 at 16:55
  • $\begingroup$ @PRD I edited my answer. See the last part! $\endgroup$ Oct 22 at 16:59
  • $\begingroup$ @Nightflight Thank you very much! $\endgroup$
    – PRD
    Oct 22 at 17:00
  • $\begingroup$ @PRD I'll say the upper/lower cases(left/right cases are exactly same way). There are 4 possibilities, $(U,L)=(1,4), (1,5), (2,4), (2,5)$. $\endgroup$ Oct 22 at 17:10

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