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# Let S be the set of sides and diagonals of a regular pentagon. A pair

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Math Expert
Joined: 02 Sep 2009
Posts: 59725
Let S be the set of sides and diagonals of a regular pentagon. A pair  [#permalink]

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20 Mar 2019, 04:45
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Difficulty:

65% (hard)

Question Stats:

40% (02:54) correct 60% (02:21) wrong based on 15 sessions

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Let S be the set of sides and diagonals of a regular pentagon. A pair of elements of S are selected at random without replacement. What is the probability that the two chosen segments have the same length?

(A) 2/5
(B) 4/9
(C) 1/2
(D) 5/9
(E) 4/5

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Math Expert
Joined: 02 Aug 2009
Posts: 8320
Let S be the set of sides and diagonals of a regular pentagon. A pair  [#permalink]

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20 Mar 2019, 05:44
Bunuel wrote:
Let S be the set of sides and diagonals of a regular pentagon. A pair of elements of S are selected at random without replacement. What is the probability that the two chosen segments have the same length?

(A) 2/5
(B) 4/9
(C) 1/2
(D) 5/9
(E) 4/5

A regular pentagon means five sided figure which has all 5 sides equal(RED in color). Now we can draw the diagonals too, and we find there are 5 diagonals ( BLACK in color).
To count total sides is to find ways to select 2 of A, B, C, D, E, that is 5C2 = 10 segments.
So choosing equal sides means choosing two of the five RED sides = 5*4, or choosing two of the five diagonals BLACK lines = 5*4
Total ways to choose 2 sides out of 10 is 10*9

Probability = $$\frac{5*4+5*4}{10*9}=\frac{4}{9}$$

B
Attachments

PENTA.png [ 70.35 KiB | Viewed 278 times ]

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Re: Let S be the set of sides and diagonals of a regular pentagon. A pair  [#permalink]

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20 Mar 2019, 07:23
Bunuel wrote:
Let S be the set of sides and diagonals of a regular pentagon. A pair of elements of S are selected at random without replacement. What is the probability that the two chosen segments have the same length?

(A) 2/5
(B) 4/9
(C) 1/2
(D) 5/9
(E) 4/5

Pentagon has 5 equal sides and 5 diagonals ; total equal sides 10
so case 1: picking 2 sides of equal length; 5/10 * 4/9 = 2/9
case 2 : picking 2 sides of equal diagonal ;5/10 * 4/9 = 2/9
IMO B
Re: Let S be the set of sides and diagonals of a regular pentagon. A pair   [#permalink] 20 Mar 2019, 07:23
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