14.5. a) Seven girls are forming a circle. In how many different ways can they stand in a circle?

b) How many necklaces can be made from seven different beads?
14.5. a) Answer: 720. Seven girls can be arranged in a circle in $7!=5040$ ways. However, in a circle, arrangements that can be obtained by rotating the circle are not considered different. By rotating the circle, one arrangement can result in 7 other arrangements. Therefore, the total number of ways is $7!/ 7=6!=720$.

b) Answer: 360. Unlike a circle, a necklace can not only be rotated but also flipped. As a result, we get $720 / 2=360$ different necklaces.