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My book has the answer to this problem but I do not understand how these answers are obtained .I know the circular permutation formula , but I cannot apply it in this specific case.

Problem: part i:Find the number of ways in which 4 boys and 4 girls can be seated around a circular table if the boys and girls are to have alternate seats.

part ii:Find the number of ways if they sit alternately around a circular table and if one boy and one girl are to sit in adjacent seats.

part iii:Find the number of ways if they sit alternately around a circular table and if one boy and one girl must not sit in adjacent seats .

The answers are

part i:3!4!=144.

part ii:2 3! 3!=72.

part iii:144-72=72.

The answers I got are way off and incorrect unfortunately....

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(i) Fix one boy. (Using the circular symmetry.) The three boys remained and the four girls can be placed independently in $3!$, respectively $4!$ ways, so all together in $3!\;4!$ ways.

(ii) Fix again the one boy, the one given in (ii). "His" girl has to sit near him, so she has two chances, left or right. The three boys remained and the three girls remained can be placed independently in $3!$, respectively $3!$ ways, so all together in $2\;3!\;3!$ ways.

(iii) Just build the difference, or copy the above and place "his" girl on one of the non-adjacent sits.


Note: (ii) has a symmetric solution obtained by letting the one specific girl first have her seat, then her boy finds/chooses easily his place, but she was checking the make-up as i was typing. (Now she is outdoor, a phone call...)

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