6 friends (Andy, Bandy, Candy, Dandy, Endy and Fandy) are out to dinner. They will be seated in a circular table (with 6 seats). Andy and Bandy want to sit next to each other to talk about the Addition Principle, Bandy and Candy want to sit next to each other to talk about the Principle of Inclusion and Exclusion. How many ways are there to seat them?

Clarification: Rotations are counted as the same seating arrangements, reflections are counted as different seating arrangements.

Question

6 friends (Andy, Bandy, Candy, Dandy, Endy and Fandy) are out to dinner. They will be seated in a circular table (with 6 seats). Andy and Bandy want to sit next to each other to talk about the Addition Principle, Bandy and Candy want to sit next to each other to talk about the Principle of Inclusion and Exclusion. How many ways are there to seat them?

Clarification: Rotations are counted as the same seating arrangements, reflections are counted as different seating arrangements.

hba · Answer

Circle
(n-1)!

Line

n!

Necklace or garland

(n-1)!/2

koikkara · Answer

720.....

koikkara · Answer

proof:

For the first seat, we have a choice of any of the 6 friends. After seating the first person, for the

second seat, we have a choice of any of the remaining 5 friends. After seating the second

person, for the third seat, we have a choice of any of the remaining 4 friends. After seating the

third person, for the fourth seat, we have a choice of any of the remaining 3 friends. After

seating the fourth person, for the fifth seat, we have a choice of any of the remaining 2 friends.

After seating the fifth person, for the sixth seat, we have a choice of only 1 of the remaining

friends. Hence, by the Rule of Product, there are 6*5*4*3*2*1=720

these 6 people.

More generally, this problem is known as a Permutation. There are |dw:1358685675757:dw|

ways to seat people in a row.