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# ABCDEFGH is an octagon. Two vertices of the octagon are selected

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Director
Joined: 16 Jan 2019
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Location: India
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ABCDEFGH is an octagon. Two vertices of the octagon are selected  [#permalink]

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19 Jun 2019, 06:29
1
5
00:00

Difficulty:

65% (hard)

Question Stats:

47% (01:42) correct 53% (02:04) wrong based on 49 sessions

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ABCDEFGH is an octagon. Two vertices of the octagon are selected at random. Find the probability that these vertices form a diagonal of the octagon.

A. $$\frac{1}{7}$$

B. $$\frac{2}{7}$$

C. $$\frac{3}{7}$$

D. $$\frac{4}{7}$$

E. $$\frac{5}{7}$$
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ABCDEFGH is an octagon. Two vertices of the octagon are selected  [#permalink]

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Updated on: 19 Jun 2019, 08:17
Number of ways you can select 2 verices= 8C2=28
Number of ways these vertices form a diagonal= nC2-n=8C2-8=20

the probability that these vertices form a diagonal of the octagon. = 20/28=5/7

firas92 wrote:
ABCDEFGH is an octagon. Two vertices of the octagon are selected at random. Find the probability that these vertices form a diagonal of the octagon.

A. $$\frac{1}{7}$$

B. $$\frac{2}{7}$$

C. $$\frac{3}{7}$$

D. $$\frac{4}{7}$$

E. $$\frac{5}{7}$$

Originally posted by nick1816 on 19 Jun 2019, 07:54.
Last edited by nick1816 on 19 Jun 2019, 08:17, edited 1 time in total.
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Re: ABCDEFGH is an octagon. Two vertices of the octagon are selected  [#permalink]

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19 Jun 2019, 08:11
2
nick1816 wrote:
Number of ways you can select 2 verices= 8C2=28
Number of ways these vertices form a diagonal= (1/2)*(8C1 * 1)=4

the probability that these vertices form a diagonal of the octagon. = 4/28=1/7

A diagonal of a polygon is any line connecting two vertices that is not an edge, so an octagon has quite a few diagonals -- I think you're only counting lines connecting vertices that are strictly opposite each other.

Here we can select a pair of vertices in 8*7/2! = 28 ways. If we draw a line between our pair of vertices, it will be a diagonal unless it's one of the 8 edges of the octagon. So we can draw 28 - 8 = 20 diagonals, and the answer is 20/28 = 5/7.
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Joined: 19 Oct 2018
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Location: India
Re: ABCDEFGH is an octagon. Two vertices of the octagon are selected  [#permalink]

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19 Jun 2019, 08:15
My bad!! Gotta edit my solution.
IanStewart wrote:
nick1816 wrote:
Number of ways you can select 2 verices= 8C2=28
Number of ways these vertices form a diagonal= (1/2)*(8C1 * 1)=4

the probability that these vertices form a diagonal of the octagon. = 4/28=1/7

A diagonal of a polygon is any line connecting two vertices that is not an edge, so an octagon has quite a few diagonals -- I think you're only counting lines connecting vertices that are strictly opposite each other.

Here we can select a pair of vertices in 8*7/2! = 28 ways. If we draw a line between our pair of vertices, it will be a diagonal unless it's one of the 8 edges of the octagon. So we can draw 28 - 8 = 20 diagonals, and the answer is 20/28 = 5/7.
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Re: ABCDEFGH is an octagon. Two vertices of the octagon are selected  [#permalink]

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21 Jun 2019, 03:23
1
The total no of diagonals for a n-sided polygon can be given by the formula- (n(n-3))/2.
so in this case total no of diagonals= 8(8-3)/2=20 diagonals

Total no. of ways of selecting 2 points out of 8= 8C2=28 ways

Thus required probability=20/28=5/7

Thus E is the right answer
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Re: ABCDEFGH is an octagon. Two vertices of the octagon are selected  [#permalink]

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09 Aug 2019, 11:39
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Octagon has 8 vertices. Probability of picking the first vertex is 8/8 (doesn't matter which one).
Then there are 7 other vertices (x/7). Can't pick the one adjacent left or adjacent right as those are not diagonal.
(7-2)/7=5/7
Re: ABCDEFGH is an octagon. Two vertices of the octagon are selected   [#permalink] 09 Aug 2019, 11:39