Calculate the number of ways in which three girls and four boys can be seated on a row of seven chairs if....

.....(Salam).....each arrangement is to be symmetrical.

please tell me how to do this question. and what does the condition mean really? that they have to be symmetrical. i'm sorry dunno that.

thank you.

Using result from the previous guy, this are the 3 symmetric ways
b b g g g b b
b g b g b g b
g b b g b b g

Now consider the three girls

g1 g2 g3, they can sit in 3 ( who is in centre) * 2 ( when one of the girl is in centre the other two could sit once on left and once of right)

Therefore g1 g2 g3 -g3 g2 g1-g2 g1 g3-g3 g1 g2 .... are 6 ways.
Similarly for 4 boys, b1 b2 b3 b4, they can sit in 4*3*2 ways = 24 ways.

As shown above there are 3 symmetric cases, and permutation for each of them gives total of

3* 24* 6 = 432 ways.

To be symmetrical in a single row, the arrangement should be the same on each side of the center.
As there is an odd number of girls, there must be a girl in the center position.

- - - g - - -

You now have to fit one girl and two boys on the left side, the right side must be the mirror image to be symmetrical.

So you have
b b g
b g b
g b b
as the only possible arrangements that meet the conditions.

Therefore, there are only three ways to do it.

b b g g g b b
b g b g b g b
g b b g b b g

The total number of combinations is 2^7 power. 2=number of genders, 7=number of seats. This is 128 possible combinations. As to symmetry I am not clear on what that means either. Sorry. I can't find symmetry in an odd number (7) anyway.

Simple play musical chairs
L O L
John