Dumbbell Attack!
How many rectangles are there in the above diagram?
The answer is 105.
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3 solutions
i counted all the rectangles and got 101. got exhausted and left he question.
If we just see one weight of the dumbbell,
Number of possible rectangles = ( 2 3 ) × ( 2 6 ) = 4 5 (Since there are three vertical lines and we have to choose two to form two sides of rectangle and there are six horizontal lines and we'll choose two for making rectangle)
Therefore, Total rectangles in both weights = 9 0
we can easily count the number of rectangles formed by the bar of dumbbell = 1 5
So, Total rectangles must be = 9 0 + 1 5 = 1 0 5 Isn't it ?
As you say, there are 45 rectangles in either dumbbell. There are also ( 2 7 ) = 2 1 rectangles contained within the middle row. However, this double-counts the rectangles that are both within a dumbbell and the middle row. There are 3 rectangles within the left dumbbell and the middle row, and 3 rectangles within the right dumbbell and the middle row, so the answer is 4 5 + 4 5 + 2 1 − 3 − 3 = 1 0 5 .
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On a second thought, you're absolutely correct. 4 of them just slipped off my sight. I thought that my answer matches with the correct answer so it must be correct. I'll change my solution to a suggestion. Thank you.
In future, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the “dot dot dot” menu in the lower right corner. You will get a more timely response that way.
Could you update your solution? Thanks!
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Thank you for telling. I was actually not familiar with the procedure of submitting a report. I've submitted a report now.
Number of rectangles in a rectangular net of dimensions A X B =
AB(A + 1)(B + 1)/4
We have 3 rectangular nets of dimensions (2X5), (2X5) , (1X6)
There are 6 rectangles in common
So
The number of rectangles = 45 + 45 + 21 - 6 = 105
Write a solution. Rectangles in “weights” = 2(3C2)(6C2) = 90.
Rectangles in “rod” = 7C2 = 21
Rectangles in “weights & rod” = 2(3C2) = 6
According to Venn’s union theory, all rectangles = 90+21-6 = 105.