How many non-degenerate rectangles are there of the form (a lessthanorequalto x

Question

How many non-degenerate rectangles are there of the form (a lessthanorequalto x lessthanorequalto b) Lambda (c lessthanorequalto y lessthanorequalto d) where a, b, c and d are integers between (and including) 0 and 99? This problem is trickier than it looks (and therefore worth a significant amount of points). You will need to think hard about your solution and very clearly explain your line of reasoning in your submitted solutions. Here are some hints Find a formula that's simple but possibly over-counts some rectangles or counts degenerates Remove the over-counts from the original formula. Remove the degenerates from the what remains Check your formula for a, b, c, d between 0 to 3 by drawing a picture and counting by hand.

Explanation / Answer

Solution:-

i) There are total 100 values of x (0 to 99) and 100 values of y (0 to 99). So, in total there are 100 * 100 = 10,000 rectangles.

ii) Now, we can simply reject the rectangles with any of the dimnesion 0. So, any rectangle with x = 0 or y = 0 will be degenerate. So, for non-degenerate rectangles, we are left with 99 values of x and 99 values of y.

Ans = 99 * 99 = 9801

iii) Now we can remove duplicate rectangles. Rectangles (1,2) and (2,1) are same. So, we need to remove these duplicates. For x = 1, there are 99 values of y.

For x = 2, there are 98 values of y.

For x = n, there are 100-n values of y.

So, summing up, we get answer = 99 + 98 + 97 + ..... + 1 = 4950

iv) So, applying our formula here, we get 3*(3+1)/2 = 6

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