Permutation

Question

A debating team consists of 5 boys and 4 girls. Find the number of ways they can sit:

1. in a round table,
2. in a row of nine chairs but the first seat is reserved for a boy,
3. in a row where just the girls are to sit together.

Answer 1

1. There are 9 children, so there are 9*8*7*6*5*4*3*2*1=362,880 permutations, but since we have a round table, and we are only considering circular arrangements we need to divide this by 9 because there are 9 ways of moving each child to the next seat without disturbing the order. That makes 40,320 permutations.
2. There are 5 boys and the chairs are in a row. We have a choice of 5 for the first chair, leaving 8 children to choose from in the next chair, so the permutations are: 5*8*7*6*5*4*3*2*1=201,600.
3. If all the girls are to sit together then the following applies: ggggbbbbb, bggggbbbb, bbggggbbb, bbbggggbb, bbbbggggb, bbbbbgggg. There are 6 ways of seating the boys and girls (b and g), and there are 24 (4*3*2*1) ways of arranging 4 girls and 120 (5*4*3*2*1) ways of arranging 5 boys, so the total number of permutation is 6*24*120=17,280.