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

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There are 7 keys in a key ring. If two more keys are to be [#permalink]  01 Aug 2008, 02:46
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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Re: probability [#permalink]  01 Aug 2008, 03:08
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

the two new keys are 1 and 2

there are 7 places for the key no 1 to go and it can go in any place. probability = 7/7

for second key, there are 8 places, and it can go only in two places, left ot right of key no 1, probability = 2/8

we can select first key in two ways, probability = 1/2
final probablity = 7/7 * 2/8 * 1/2 = 1/8 answer ...

EDIT : highlighted step is not required. the answer should be : 7/7 * 2/8 = 1/4

Last edited by durgesh79 on 01 Aug 2008, 06:20, edited 1 time in total.
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Re: probability [#permalink]  01 Aug 2008, 03:14
durgesh!!! good explanation..
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Re: probability [#permalink]  01 Aug 2008, 03:21
thank durgesh79
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Re: probability [#permalink]  01 Aug 2008, 04:48
Just one question. Why do we need to multiple by 1/2. Can you please explain??
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Re: probability [#permalink]  01 Aug 2008, 05:31
Can you explain what is wrong with this approach ? I don't get it...

There are 7 keys on a ring, so 7 places where the keys can go to be adjacent.

You can place each of the additional 2 keys in any of these places : total number of possibilities is 7*7=49

But to have them adjacent, you have to place them in the same place : 7 possibilities

==> probability of having them adjacent is 7/49 = 1/7
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Re: probability [#permalink]  01 Aug 2008, 05:39
Oski wrote:
Can you explain what is wrong with this approach ? I don't get it...

There are 7 keys on a ring, so 7 places where the keys can go to be adjacent.

You can place each of the additional 2 keys in any of these places : total number of possibilities is 7*7=49

But to have them adjacent, you have to place them in the same place : 7 possibilities

==> probability of having them adjacent is 7/49 = 1/7

The number of possibilities is 7*8 = 56

Once you add the first key, they are now 8 => 8 possible places for the second key to go.
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Re: probability [#permalink]  01 Aug 2008, 05:43
Nerdboy wrote:
The number of possibilities is 7*8 = 56

Once you add the first key, they are now 8 => 8 possible places for the second key to go.

Why do you insert one key before the other ? I do not see it written anywhere.

What I say is that the two possibilities (left and right of the first inserted key) could just as well be counted as one.

There are 7 places where to place the 2 keys, it is not by dividing a place by 2 (with your first key) that you should be able to change the probability of the second key.
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Re: probability [#permalink]  01 Aug 2008, 06:01
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

The answer is D. I will explain if needed.
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Re: probability [#permalink]  01 Aug 2008, 06:18
zoltan wrote:
Just one question. Why do we need to multiple by 1/2. Can you please explain??

now when you asked, i realized that we dont have to multiply with 1/2. The answer should be 1/4 Option D...

Thanks for correcting me....
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Re: probability [#permalink]  01 Aug 2008, 06:22
@ maratikus: if you have a different approach, please tell me details
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Re: probability [#permalink]  01 Aug 2008, 06:25
another method

number of ways 9 keys can be arranged in a ring = (9-1)! = 8!
if we consider two keys as a unit, number of ways 8 keys can be arranged = (8-1)! = 7!
these two keys can interchange their postions in 2 ways

required probability = 2*7! / 8! = 1/4
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Re: probability [#permalink]  01 Aug 2008, 06:37
fiesta wrote:
@ maratikus: if you have a different approach, please tell me details

There are 7 ways of putting two keys next to each other -> put both of them in front of key1, key2, ..., key7. There are 28 ways of adding two keys. 7 places for key new1, 8 places for key new2 (order doesn't matter -> divide by 2). 7/28 = 1/4 -> D
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Re: probability [#permalink]  01 Aug 2008, 06:42
durgesh79 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

the two new keys are 1 and 2

there are 7 places for the key no 1 to go and it can go in any place. probability = 7/7

for second key, there are 8 places, and it can go only in two places, left ot right of key no 1, probability = 2/8

we can select first key in two ways, probability = 1/2
final probablity = 7/7 * 2/8 * 1/2 = 1/8 answer ...

EDIT : highlighted step is not required. the answer should be : 7/7 * 2/8 = 1/4

Hi durgesh,

Could you explain why the answer below is wrong?
1 x 2/8
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Re: probability [#permalink]  01 Aug 2008, 06:45
judokan wrote:
Hi durgesh,

Could you explain why the answer below is wrong?
1 x 2/8

i think 1/4 is the right answer.... earlier i came up with 1/8 but later realised my mistake...
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Re: probability [#permalink]  01 Aug 2008, 06:59
durgesh79 wrote:
another method

number of ways 9 keys can be arranged in a ring = (9-1)! = 8!
if we consider two keys as a unit, number of ways 8 keys can be arranged = (8-1)! = 7!
these two keys can interchange their postions in 2 ways

required probability = 2*7! / 8! = 1/4

At first, I had the same approach which I often use, but I still thought 9 keys arranged in a line >>> the number of ways can be arranged in a line = 9!
the number of ways 2 keys can be arranged adjacent = 2!*8!
and the answer = 2!*8!/9! = 2/9
So I was confused, I only guested 1/8
Hi, thank you, durgesh79, here is a ring, not a line
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Re: probability [#permalink]  01 Aug 2008, 07:07
maratikus wrote:
fiesta wrote:
@ maratikus: if you have a different approach, please tell me details

There are 7 ways of putting two keys next to each other -> put both of them in front of key1, key2, ..., key7. There are 28 ways of adding two keys. 7 places for key new1, 8 places for key new2 (order doesn't matter -> divide by 2). 7/28 = 1/4 -> D

Thanks maratikus, I try to understand, but the explanation is not clear. Please again
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Re: probability [#permalink]  01 Aug 2008, 07:41
arjtryarjtry wrote:
durgesh!!! good explanation..

arjtryarjtry !! I'm curious to know the OA... Does that reaction of yours mean OA is 1/8
Could you please confirm the OA, becuase now we are getting 1/4
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Re: probability [#permalink]  01 Aug 2008, 07:45
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

IMO D)

Probability of two keys to be adjacent = Probability when 2nd key is added it is adjacent to 1st key added = 2/Number of spaces = 2/8 = 1/4
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Re: probability [#permalink]  01 Aug 2008, 08:28
I get D, 1/4. Here is my reason why.

I think the answer makes us assume that you place the keys onto the ring 1 at a time. We start with 7, and add 1. This one doesn't matter where it goes so it is not factored into the probability.

The second key does matter because we need to figure out where it can go and what is the chance it will be next to (means on either side) of the key we just put on the ring. We now have 8 keys on the ring.

We must place the 2nd key into a space between the keys, so we actually have 9 spaces. Look below: (let the O represent an existing key on the ring).

|O|O|O|O|O|O|O|O|

However, this is not entirely correct because the questions states the keys are on a ring, not laid out on a flat surface it appears to be above with the |O|. So the first and last | is actually the same because the ring is circular. This gives us 8 spots the key can go. If the new key (first one we placed) is represented by O, then there are 2 places that will satisfy the question of the keys being placed together. That is 2 places out of 8. so 2/8 probability reduces to 1/4 (D).

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

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Re: probability   [#permalink] 01 Aug 2008, 08:28

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