(a) In how many ways can 8 different beads be strung on a necklace? (b) In how many ways can 8 boys from a ring

Solution

Let's a necklace is circular, so fix position of 1 Bead then 7 beads can be arranged in 7! ways.

Now it is a necklace, so the clockwise and anti-clockwise arrangement must be the same. So the required number formed = (1/2)x7! = (1/2)x5040 = 2520

Here 8 boys form a ring, so fix position of 1 boy then 7 boys can be arranged in 7! = 5040 ways.