P1. (30) Twelve students (six girlfriend-boyfriend pairs) attend a dance party w

Question

P1. (30) Twelve students (six girlfriend-boyfriend pairs) attend a dance party which involves holding hands to form a circle. (a) (10) How many possible configurations are there if there are no re- (b) (10) How many possible configurations are there if boys and girls strictions about who is next to whom? must alternate, but any boy can be next to any girl? (c) (10) How many possible configurations are there if boys and girls must alternate, but each girl must be next to her boyfriend in the circle?

Explanation / Answer

Question P1

here there are twleve (students) or 6 couple

(a) Number of possible configurations when it doesn't matter who is next to whom, it can be imagines that if we fix one boy at one point, there are 11 possitions to fill so there are 11! combinations to fill these 11 spaces= (12-1)! = 11!

(b) Here if boys and girls must alternate, but any boy can be next to any girl. That mean if we fix a boy at one place then it means that 6 places will be fixed where all girls can occupy. Similarly, for boys, there are 5 places where the other 5 boys can occupy the places.

So different combinations = 6! * 5! = 86400

(c) Here each girl must be opposite to her boyfriend in the circle. So here is we fix a boy to certain position, that means her girfriend position will also be fixed. So, now the positions of remaining 5 boys are also fixed but they can interchange so there are 5! combinations. All of their girlfrinds will occupy their places opposite to their boyfriends.

so total such combinations = 5! = 120

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