At a circus, there is a large circular carousel with ten seats. Six pe

Question

At a circus, there is a large circular carousel with ten seats. Six performers dressed as clowns and four performers dressed as acrobats are getting ready to sit on the carousel for a grand act. However, the ringmaster has one strict rule: no two acrobats can sit next to each other during the performance. How many distinct seating arrangements can they create under this rule?

(A) 14,400
(B) 43,200
(C) 51,200
(D) 72,000
(E) 86,400

Show Answer

Official Answer

B

(A) 14,400
(B) 43,200
(C) 51,200
(D) 72,000
(E) 86,400

Krunaal · Answer

6 Clowns can be arranged in circular carousel in (6-1)! = 5! ways =  120

Now for the remaining 4 acrobats there are 6 places vacant between clowns, selecting 4 places = 6C4 =  15

4 acrobats can be arranged in 4! ways =  24

120*15*24 =  43,200

Answer B.

RandomGintoki · Answer

hey why did you take 6c4 ? Should the answer not be 4!*120 as after 6 clowns have been seated only 4 other chair remain

Krunaal · Answer

If we directly take 4! we are allowing arrangements where acrobats can sit together.

So when we sit 6 clowns in a circle, there are 6 distinct gaps around the circle => C1 _ C2 _ C3 _ C4 _ C5 _ C6 and  one  b/w C1 and C6 as they're sitting in a circle. Now acrobats need to be seated in these gaps, so that none of them sits together and there are 6 gaps, so we first choose 4 in 6C4 ways and then 4 acrobats can sit in 4! ways.

bumpbot · Answer

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At a circus, there is a large circular carousel with ten seats. Six pe

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At a circus, there is a large circular carousel with ten seats. Six pe

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