Circular Chain

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sk8ternite: Master | Next Rank: 500 Posts; Posts: 119; Joined: Sun May 10, 2009 7:46 pmFollowed by: 1 member; Thanked: 3 times

by sk8ternite » Wed Aug 26, 2009 1:50 pm

A circular chain consists of 4 gems: a diamond, an emerald, a ruby, and a sapphire. In how many ways can the gems be arranged on the chain?

a. 2
b. 3
c. 6
d. 108
e. 116

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Source: Beat The GMAT — Problem Solving | Wed Aug 26, 2009 1:50 pm

vani_13in: Junior | Next Rank: 30 Posts; Posts: 21; Joined: Tue Aug 19, 2008 4:58 am

by vani_13in » Wed Aug 26, 2009 3:07 pm

I think (n-1)! i.e. 4-1=3! Ans= 6

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sk8ternite: Master | Next Rank: 500 Posts; Posts: 119; Joined: Sun May 10, 2009 7:46 pmFollowed by: 1 member; Thanked: 3 times

by sk8ternite » Wed Aug 26, 2009 3:20 pm

vani_13in wrote:I think (n-1)! i.e. 4-1=3! Ans= 6

(n-1)!, why did you choose this formula and can you explain the background behind this formula?

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

by Stuart@KaplanGMAT » Wed Aug 26, 2009 9:27 pm

sk8ternite wrote:
vani_13in wrote:I think (n-1)! i.e. 4-1=3! Ans= 6
(n-1)!, why did you choose this formula and can you explain the background behind this formula?

That's the general formula for arranging n distinct objects in a circle.

The reason why a circle has fewer permutations than a straight line is because there are n duplicate scenarios for every unique arrangement, depending on where you put the first object.

For example, let's say we're arranging 6 people, A, B, C, D, E and F, around a table. We don't care where they sit, but let's call the chairs 1, 2, 3, 4, 5 and 6.

One arrangement would be:

A1, B2, C3, D4, E5 and F6.

Another arrangement would be:

A2, B3, C4, D5, E6 and F1.

However, looking at the people relative to each other, these two arrangements are identical. So, if we don't factor out n, we have multiple counting.

You could look at the formula as:

n!/n, which is of course the same as (n-1)!

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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praky_rules: Senior | Next Rank: 100 Posts; Posts: 45; Joined: Wed Jun 11, 2008 4:25 pm; Thanked: 4 times; GMAT Score:730

by praky_rules » Sat Aug 29, 2009 4:53 pm

Answer should n-1!/2 = 3 as the chain can be viewed from either side(front and back!)

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