Question

I don't understand the difference between these two problems. Problem #1 has you ₄ P ₄ / 4 = 4!/4 because I guess you can rotate it 4 times and it'd still be in the same arrangements as each of those women still have the same neighbors during those 4 rotations. Please correct me if I am wrong.

However, in Problem #2, this time has 3 boys and 4 girls sitting at a circular table, to which boys have to sit with boys and girls have to sit with girls. Here you would have a permutation for the boys and a permutation for the girls. ALTHOUGH, why is it that you don't divide the permutations here by "n" as you did in Problem #1. Aren't the arrangements still the same and you don't want to count some of those?

1) Find the number of different ways of arranging 4 women around a circular conference table

₄ P ₄ / 4 = 4!/4 = 3 * 2 * 1 = 6

2) How many ways could the three boys and four girls be arranged around a circular table if the boys must sit together and the girls as well?

Permutation of Boys * Permutations of Girls

= ₃ P ₃ * ₄ P ₄

= 3!/0! * 4!/0!

= 3 * 2 * 1 * 4 * 3 * 2 * 1

= 144 <-- Why don't you divide _n P _r by n here as we did in Problem #1?

Answer 1

2) We will take the three boys into one group and the four girls also into one group. So there will be total two group.

So the no of ways = 2P2/2 * 3P3 * 4P4

                              = 2!/2 * 3!/0! * 4!/0!

                              = 1 * 3 * 2 * 1 * 4 * 3 * 2 * 1

                              = 144