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Exam-Style Question on Combinatorics

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Question id: 636. This question is similar to one that appeared on an IB AA Higher paper in 2022. The use of a calculator is allowed.

Fifty years ago there was a zoo where animals were housed in cages. There were eight animal cages in a line on the northern perimeter of the zoo. Only four cages were being used: one to hold the gorilla, one for the chimpanzees, one for the gibbons and one for the macaques.

(a) Find the number of ways the four cages can be assigned to the animals so that the cages are next to each other.

It was decided that the gorilla should have at least one empty cage between its cage and the next occupied cage. The other animal cages do not have to be next to each other.

(b) Find the number of ways the four cages can be assigned now.


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