# Thread: Discrete Math Problem! HELP!!

1. ## Discrete Math Problem! HELP!!

Let m,n be positive integers. An m x n chessboard is a rectangle of length m and height n that is divided into mn squares of length 1. For example, a stardard chessboard is 8 x 8. A rectangle on a chessboard, is a rectangle all of whose sides are made up of sides of the 1 x 1 squares.

How many rectangles are there on an m x n chessboard? For example, when m = n = 2, then the answer is 9.

I just don't know where to begin! Any help is greatly appreciated. Thanks.

2. You are able to put numbers standing by each vertical and horizontal line of the chessboard. Just like what I am doing below,
[HTML]
1 2 3
--- ---
| | |
2 --- ---
| | |
3 --- ---
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Then I can say for the longer border of rectangle, I have 3 pairs of start and end points, since picking up 2 numbers from 3 numbers has 3 possibilities, they are 12, 13, 23.

Similarly, I also have 3 choices when facing the situation of the shorter border of the rectangle. The possible combinations are 12, 13, 23.

So, for the rectangle permutation, I have 9 choices, they are 12-12, 12-13, 12-23, 13-12, 13-13, 13-23, 23-12, 23-13, 23-23. They are exactly the answers to your example question.

For general cases, just think about it in the same way, then you may come to the conclusion that
[HTML]
-- 2 -- 2 m! n!
Answer = | x | = ------------------------
-- m -- n 2！ 2！ (m-2)! (n-2)!
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