The five sides of a pentagon have lengths of 2, 3, 4, 5 and 6 inches.

Question

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

GMATinsight · Accepted Answer

CONCEPT : The number of ways to arrange n elements around a circular Manner is (n-1)!.

But the number of ways to arrange n elements in circular manner where the arrangement can be flipped around (e.g. Necklace, Garland, Bracelet) = (n-1)!/2.

The pentagon in the problem above can be flipped over (Clockwise Arrangement of Sides = Counter-clockwise arrangement), the number of distinct ways to arrange the 5 sides = (5-1)!/2 = 12.

Answer: option B

broall · Answer

Tricky one. :-D

5 sides of that pentagon are $a,b,c,d,e$ The totale numbers of possibles pentagons are $5 \times 4 \times 3 \times 2 \times 1 =5!$ Note that each possible selection of $(a,b,c,d,e)$, there are 4 other possible selections that are formed by arranged around each element. For example these 5 selections are considered the same pentagons $\begin{split} 2,3,4,5,6\ 6,2,3,4,5\ 5,6,2,3,4\ 4,5,6,2,3\ 3,4,5,6,2\ \end{split}$ Hence, in order to count the possible different selections, we need to divide the result by 5: $5!/5=4!$ Note that each pentagon could be fliped over, so both are considered the same. For example, these 2 selections are considered the same pentagons $\begin{split} 2,3,4,5,6\ 6,5,4,3,2\ \end{split}$ So, we also need to divide the result by 2 to avoid flipped over ones: $4!/2=12$ The final result are 12. The answer is B. If the result are 24, this means that it covers the flipped over pentagons.

ShashankDave · Answer

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.

Fibo1618 · Answer

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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The five sides of a pentagon have lengths of 2, 3, 4, 5 and 6 inches.

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The five sides of a pentagon have lengths of 2, 3, 4, 5 and 6 inches.

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