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Worked solution

Seat eight distinct people round a table

In how many ways can 4 boys and 4 girls be seated at a round table if (i) there is no restriction, (ii) the boys and the girls must be separated, (iii) one particular girl must be seated between 2 particular boys?

CJC · Lecture · Example 18

(i) Remove rotations of the same seating

The eight people are distinct and seats are unlabelled. Fix one person as a reference and arrange the other seven: (8−1)!=5040(8-1)!=5040.

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(ii) Alternate boys and girls using circular gaps

The source interprets “boys and girls must be separated” as alternating genders. Arrange the four boys around the table in (4−1)!(4-1)! ways. Their four intervening gaps each receive one girl, in 4! ways. Thus (4−1)! 4!=144(4-1)!\,4!=144. Unlike a row there are four gaps, not five.

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(iii) Make a consecutive boy-girl-boy block

Group the particular people as B1,G1,B2, with the girl immediately between the boys. Together with the other five people this gives six outer objects, arranged circularly in (6−1)!(6-1)! ways. Swap the two boys at the ends in 2! ways. Total (6−1)!×2!=240(6-1)!\times2!=240.

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Source part (ii) has four boys round a circle with an arrow into each of the four intervening gaps. Place one girl in each gap. Part (iii) has a consecutive block B1,G1,B2 plus five remaining people around the circle: six outer objects. Reverse the two boys within the block in two ways.
Source part (ii) has four boys round a circle with an arrow into each of the four intervening gaps. Place one girl in each gap. Part (iii) has a consecutive block B1,G1,B2 plus five remaining people around the circle: six outer objects. Reverse the two boys within the block in two ways.
Source part (ii) has four boys round a circle with an arrow into each of the four intervening gaps. Place one girl in each gap. Part (iii) has a consecutive block B1,G1,B2 plus five remaining people around the circle: six outer objects. Reverse the two boys within the block in two ways.

Common mistakes

Rotations are identified but reflections are distinct. Part (ii) follows the source alternating-gender interpretation; part (iii) requires immediate neighbours, represented by one contiguous block.

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