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Circular Permutation

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Circular Permutation

by ttd » Sat Apr 03, 2010 2:19 pm
How many different ways can you seat 10 men and 10 women around a circular table if you need to alternate the men and women?

A) 100
B) 9! x 9!
C) 20! / 10!
D) 9! x 10!
E) 10! x 10!

-- Please explain why the answer is D, not B. Thanks!
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by targetperfect » Sat Apr 03, 2010 6:16 pm
ttd,
It works like this... before getting to circular permutation, think about linear permutation. Say there are 10 Men and 10 Women you want to seat them alternatively....
*M*M*M.....M*
you know you would like to place the Women in the * place holders.... how many * you got ? 11 right ? So it works like this

n! * (n+1)Pt as the total no. of permutations..

n - no. of men
t - no. of women

so it is

10! * (10+1)P10 which is 10! * 11!

Now lets talk about the circular combination.. you know how to arrange n objects in a circle, there are (n-1)! possible permutations.. So basically in a circular setting the no. of place holders go down by 1. So replace n with n-1 in the above formula
you get
(n-1)! * (n-1+1)Pt

(10-1) * (10-1+1)P10

Where n=10 = t

Hence 9! * 10!
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by eaakbari » Sat Apr 03, 2010 10:37 pm
For circular permutations, to arrange n elements , the formula is (n-1)!

Now consider 10 men
They can be arranged in 10-1= 9! ways

The 10 women can be put in between two men in exactly 10! ways

Hence 9! * 10!

IMO D
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