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Circular Probability


manasdas

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1. In how many ways can five people sit around a circular table if one person can not have the same neighbors in any two

arrangements?

 

2. Seven men and seven women have to sit around a circular table so that no 2 women are together. how many ways?

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1. In how many ways can five people sit around a circular table if one person can not have the same neighbors in any two

arrangements?

 

If there was no restriction 5 people can sit around a circular table in (5-1)! = 4! = 24 ways

 

If you try to visualize this in a circle, you'll understand that out of these arrangements, half would be counter-clockwise equivalent to the other half ... Considering only one direction we have 24/2 = 12 ways.

 

 

2. Seven men and seven women have to sit around a circular table so that no 2 women are together. how many ways?

 

Arranging 7 men around a circular table = 6! ways

 

There would be 7 spots in between any two men in the above arrangments and the 7 women can sit in any of those 7 places in 7! ways (note: this is not a circular arrangement)

 

So, the total ways = 6!*7!

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  • 3 years later...
If there was no restriction 5 people can sit around a circular table in (5-1)! = 4! = 24 ways

 

If you try to visualize this in a circle, you'll understand that out of these arrangements, half would be counter-clockwise equivalent to the other half ... Considering only one direction we have 24/2 = 12 ways.

 

 

Arranging 7 men around a circular table = 6! ways

 

There would be 7 spots in between any two men in the above arrangments and the 7 women can sit in any of those 7 places in 7! ways (note: this is not a circular arrangement)

 

So, the total ways = 6!*7!

 

For Prob-1:

how about the following circular arrangements: (In all four arrangements C has the same two neighbours - B & D)

1) ABCDE

2) EBCDA

3) EDCBA (Anti clockwise of 1)

4) ADCBE (Anti clockwise of 2)

 

Shouldn't we be dividing (5-1)! by 4?

 

For Prob-2:

Why is the arrangement of women not considered circular?

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