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# There are 7 keys in a key ring. If two more keys are to be

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Current Student
Joined: 11 May 2008
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There are 7 keys in a key ring. If two more keys are to be [#permalink]

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05 Sep 2008, 08:39
This topic is locked. If you want to discuss this question please re-post it in the respective forum.

There are 7 keys in a key ring. If two more keys are to be added in the ring at random, what is the probability that two keys will be adjacent?
(A) 1/8
(B) 1/6
(C) 1/5
(D) 1/4
(E) 1/3

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Director
Joined: 23 Sep 2007
Posts: 782

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Re: ring probability [#permalink]

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05 Sep 2008, 09:28
arjtryarjtry wrote:
There are 7 keys in a key ring. If two more keys are to be added in the ring at random, what is the probability that two keys will be adjacent?
(A) 1/8
(B) 1/6
(C) 1/5
(D) 1/4
(E) 1/3

D

7 keys in the ring, if you add 1 more, you will have 8 keys. The ring is circular, so you have 8 slots for putting in the next key, but the next key can be to the left or to the right of the 8th key. so 2/8=1/4

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Manager
Joined: 15 Jul 2008
Posts: 206

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Re: ring probability [#permalink]

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05 Sep 2008, 09:30
arjtryarjtry wrote:
There are 7 keys in a key ring. If two more keys are to be added in the ring at random, what is the probability that two keys will be adjacent?
(A) 1/8
(B) 1/6
(C) 1/5
(D) 1/4
(E) 1/3

1/4.

Total number of ways to arrange 9 keys in circular permutation = 8!

Consider the 2 keys in question as unit. So you have 8 units which can be arranged in 7! ways. The pair can be arranged in 2 ways. So total ways in which the pair always stays together = 2*7!

Prob = (2*7!) / 8! = 1/4

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VP
Joined: 05 Jul 2008
Posts: 1403

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Re: ring probability [#permalink]

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05 Sep 2008, 15:21
number of ways for arranging n people in a circle is (n-1)!

Total 9 keys in a ring 8! ways

Adjacent means keep the two rings as one unit and the 8 rings can be arranged in 7! ways. the 2 rings can be arranged in 2 ways in between them

Ans is ( 2X 7!) / 8! which is 1/4

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Posts: 1795

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Location: New York
Re: ring probability [#permalink]

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05 Sep 2008, 17:32
Agree with D..
both logics are good.
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Manager
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Schools: ISB, Tuck, Michigan (Ross), Darden, MBS
Re: ring probability [#permalink]

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06 Sep 2008, 00:26
bhushangiri wrote:
arjtryarjtry wrote:
There are 7 keys in a key ring. If two more keys are to be added in the ring at random, what is the probability that two keys will be adjacent?
(A) 1/8
(B) 1/6
(C) 1/5
(D) 1/4
(E) 1/3

1/4.

Total number of ways to arrange 9 keys in circular permutation = 8!

Consider the 2 keys in question as unit. So you have 8 units which can be arranged in 7! ways. The pair can be arranged in 2 ways. So total ways in which the pair always stays together = 2*7!

Prob = (2*7!) / 8! = 1/4

Nice explanation!
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Re: ring probability   [#permalink] 06 Sep 2008, 00:26
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# There are 7 keys in a key ring. If two more keys are to be

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