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honchos
The five sides of a pentagon have lengths of 2, 3, 4, 5 and 6 inches. Two pentagons are considered different only when the positions of the side lengths are different relative to each others. What is the total number of different possible pentagons that could be drawn using these five side lengths ?
(A) 5
(B) 12
(C) 24
(D) 32
(E) 120

Source: Jeff Sackman

I am confident that the OA is wrong. The answer must be 24. Flipping over, or counting both clockwise and counter-clockwise arrangements the same is true only in case of necklaces, as they can be actually flipped over and worn. In this case of Drawing a pentagon on the paper, flipping over will result into nothing. When flipped over, you will get a plain paper with nothing. What I mean to say is that here the clockwise and counter-clockwise orientations definitely differ.
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honchos
The five sides of a pentagon have lengths of 2, 3, 4, 5 and 6 inches. Two pentagons are considered different only when the positions of the side lengths are different relative to each others. What is the total number of different possible pentagons that could be drawn using these five side lengths ?
(A) 5
(B) 12
(C) 24
(D) 32
(E) 120

Source: Jeff Sackman

I am confident that the OA is wrong. The answer must be 24. Flipping over, or counting both clockwise and counter-clockwise arrangements the same is true only in case of necklaces, as they can be actually flipped over and worn. In this case of Drawing a pentagon on the paper, flipping over will result into nothing. When flipped over, you will get a plain paper with nothing. What I mean to say is that here the clockwise and counter-clockwise orientations definitely differ.

Actually, if you flip the pentagon, you get its reflection. To simplify things, try to imagine 2 equal triangles looking like the photo attached.

They might look different since one is the flipped version of the other, but they are equal, and should be counted as one. Same for the pentagons. If you flip them, basically you created the same thing from a different perspective, that doesn't change the fact that they are the same. Therefore, among these 24 results, 12 are just the flipped version and should not be counted.
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reflect-graph.gif
reflect-graph.gif [ 2.86 KiB | Viewed 5064 times ]

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