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Carla is having a dinner party and is inviting 5 guests, 3 [#permalink]
19 Sep 2005, 13:51

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9. Carla is having a dinner party and is inviting 5 guests, 3 boys and 2 girls. Carla knows that she needs to sit at the head of the table, but hasnâ€™t decided on fixed seats for anyone else. If she decides not to sit any of her guests next to anyone else of the same gender, how many different seating arrangements are possible?

I don't think it's circular because there is no "head" of the table with a circular table. Plus, wouldn't the question say that the table is circular since the formula is different for circular combos? I admit that the question needs further definition.

carla at the head of the table
2 boys sit beside her chair
2 girls sit on the chairs next to the boys
and finally one guy sits on the other head chair
this way no person of same sex sitting beside each other
so we get 3! arrangements for boys and 2! arrangments for girls
total 12

Moving clockwise around the table, Carla is at the head of the table and we know that seat 1 must be occupied by a boy, and there are 3 different boys who could fill this spot. Seat 2 must then be a girl. There are 2 different girls who could fill this spot. Seat 3 is another boy. Now that one of the boys is seated (next to Carla), only 2 boys remain for this seat. Seat 4 is a girl, and only one girl remains to sit here. Finally, only one boy remains for seat 5 next to Carla. So we have 3x2x2x1x1 = 12

Moving clockwise around the table, Carla is at the head of the table and we know that seat 1 must be occupied by a boy, and there are 3 different boys who could fill this spot. Seat 2 must then be a girl. There are 2 different girls who could fill this spot. Seat 3 is another boy. Now that one of the boys is seated (next to Carla), only 2 boys remain for this seat. Seat 4 is a girl, and only one girl remains to sit here. Finally, only one boy remains for seat 5 next to Carla. So we have 3x2x2x1x1 = 12

Although I assumed while solving this question, how do we know if the table is round or rectangular..what is head of the table in circular table? does ans differ?

I think that what is meant by this question is a rectangular table.

Head of table at one end, two seats on either side of that end, one seat at opposite end to head of table. No difference between this mathematically and a circular table with one fixed seat.

In a circular arrangement, head of table is defined as wherever the head of the table wants to sit !