Probability Question

[abhijeetsinghai]

Six boys and six girls sit slong s line alternatively in x ways and along a circle in y ways(again alternatively) , then find the relation between x and y .

a. x = 12y
b. x = 2y
c. x = 6y
d. x= 8y

OA:[spoiler]a

please tell me the approach.thanx !![/spoiler]

[Uri]

Scenario I: Straight line arrangement
6 boys can sit in 6 seats in 6! ways
6 girls can sit in 6 seats in 6! ways
Either of the groups can take the first seat.
So, the total no. of possible arrangements is 2*6!*6!
This is x.

Scenario II: Circular arrangement
Let us first offer the sit to a girl! Once this girl sits, the rest of the girls can sit in 5! ways, keeping one seat vacant between each one of them.
Once the girls are seated, the boys can be arranged in 6 seats in 6! ways.
Since this is a circular arrangement, we don't have to consider whether the boy sits first or the girl.
So, the number of arrangements= 5! * 6!
This is y.

Therefore x/y= 12
or x=12y

Can anyone verify whether the above explanation is correct? I am very weak in Permutation & Combination. But still, somehow gathered enough courage to try this problem Please help :!:

[ontopofit]

yes it's correct...besides there is direct formula that number of permutations in circular arr. is (n-1)!.....i.e in this case y=11! and x=12!.hence x=12y.

however, i appreciate the way u have dun it. its better not to depend too much on direct formulas.

[raunekk]

hi uri..
i guess the explanation was perfect...

[vittalgmat]

yes it's correct...besides there is direct formula that number of permutations in circular arr. is (n-1)!.....i.e in this case y=11! and x=12!.hence x=12y.

however, i appreciate the way u have dun it. its better not to depend too much on direct formulas.

Some one pls correct me if I am wrong.
I think ontopofit's approach is incorrect, but since we are looking at a ratio, one can arrive at the same answer as long the same method is used to obtain the value of x and y.
I think Uri's approach is correct.