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A key ring has 7 keys. How many different ways can they be a [#permalink]

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30 Sep 2012, 17:27

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A key ring has 7 keys. How many different ways can they be arranged?

A. 6 B. 7 C. 5! D. 6! E. 7!

I would like to argue against OA. 7 => 6! circular permutations. Therefore, for a keyring - arrangements = 6!/2 (clockwise/anti-clockwise doesn't matter)

Re: A key ring has 7 keys. How many different ways can they be a [#permalink]

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01 Oct 2012, 03:18

voodoochild wrote:

A key ring has 7 keys. How many different ways can they be arranged?

a 6 b 7 c 5! d 6! e 7!

I would like to argue against OA. 7 => 6! circular permutations. Therefore, for a keyring - arrangements = 6!/2 (clockwise/anti-clockwise doesn't matter)

Hi voodoochild,

Why wouldn't the direction matter? It's a circular arrangement and you will look at the keyring from one side. Say there are three rings, A, B and C.. .ABC and CBA are two different arrangements, even in a circular arrangement.
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Re: A key ring has 7 keys. How many different ways can they be a [#permalink]

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01 Oct 2012, 03:42

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voodoochild wrote:

A key ring has 7 keys. How many different ways can they be arranged?

a 6 b 7 c 5! d 6! e 7!

I would like to argue against OA. 7 => 6! circular permutations. Therefore, for a keyring - arrangements = 6!/2 (clockwise/anti-clockwise doesn't matter)

Think of choosing one specific key and placing it on the key ring. The remaining 6 keys can now be arranged in 6! ways. Once you have placed one key, you created a row (straighten the circle), so the regular rule for permutations apply. It doesn't matter if you slide the keys clockwise or counter clockwise, they have the same relative position each to the other. For example, just for 3 keys, ABC and ACB are different arrangements. Mirror image arrangements are different, because you don't flip your key ring. At least I think this is what is assumed.

In the case of arrangements around a round table, it is obvious. For sure, you cannot flip your table and the people with it.
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

I read this concept while preparing for Indian Engineering entrance examination.... not sure....

Necklace, with spherical beads, can be flipped, so symmetrical arrangements can be considered identical. But, if not explicitly stated, I think you shouldn't assume anything like this.
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PhD in Applied Mathematics Love GMAT Quant questions and running.

Re: A key ring has 7 keys. How many different ways can they be a [#permalink]

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01 Dec 2013, 06:52

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: A key ring has 7 keys. How many different ways can they be a [#permalink]

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01 Dec 2013, 23:32

This question was ambiguous so I got it wrong. I interpret "keys on a ring" meaning the relative ordering can't change, only moved around the ring. That means either 1 ordering (not an option here), or something where I consider the ordering left-to-right as the keys hang on the ring, but I can move the leftmost key around the loop to put it in the rightmost position, as follows:

Re: A key ring has 7 keys. How many different ways can they be a [#permalink]

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16 Oct 2016, 10:51

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
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Re: A key ring has 7 keys. How many different ways can they be a [#permalink]

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08 Mar 2017, 23:08

voodoochild wrote:

A key ring has 7 keys. How many different ways can they be arranged?

A. 6 B. 7 C. 5! D. 6! E. 7!

I would like to argue against OA. 7 => 6! circular permutations. Therefore, for a keyring - arrangements = 6!/2 (clockwise/anti-clockwise doesn't matter)

In a permutations problem that involves a circular arrangement (such as keys on a ring, or people seated around a circular table), the number of permutations is equal to (N - 1)!. This is because there is no beginning or end to the ring (or no left end or right end). Consider the three-item arrangement of A, B, C. In a row, ABC is different from BCA, but if they're in a circle they're the same arrangement: B is between A and C, A is between C and B, etc.

So in this problem, with 7 keys, the calculation is (7 - 1)!, which is 6! which equals 720.
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