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Six boys and six girls sit along a line alternately in $x$ ways and then sit along a circle alternately is $y$ ways. then express a relationship between $x$ and $y$ .

The answer is supposed to be $x=12y$ however I am having trouble understanding it. Here is how i attempted it. Please correct me.

For the linear arrangement , let the six boys first occupy six seats leaving a seat each seat between any two boys vacant. They can be arranged in $6!$ ways. Now there are $7$ seats for the girls so the girls can be arranged in ${{7}\choose{6}}6!$ ways. So the linear arrangement of boys and girls is possible in $(7!)(6!)$ ways. For the circular arrangement, place a boy first and arrange the remaining boys in alternating seats in $5!$ ways. Now , similarly the girls can be arranged in $5!$ ways. Hence the circular arrangement is possible in $(5!)(5!)$ ways. However , according to the solution that I read , the the linear arrangement is possible in $2(6!)^2$ and the circular arrangement is possible in $(5!)(6!)$ways. Can you please explain where I went wrong ? Thank you for your help !

Aditi
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1 Answers1

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The simple way is to bend the line into a circle. The only change is we don't know where the line started, which is a factor $12$ and we are done.

In your approach you don't have seven seats for girls, you have six seats plus a factor $2$ for whether the first seat is a girl or boy. There are $2\cdot 6! \cdot 6!$ ways to arrange them in a line. The $5!6!$ for a circle is correct and the ration between these is again $12$

Ross Millikan
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