Your question was

13 (c) In how many ways can one yellow, two red and four green beads be placed on a bracelet if the beads are identical apart from colour? [Hint : This will require a listing of patterns to see if they are identical when turned over.]

To arrange 1 yellow, 2 red, and 4 green objects in a line, the number of possible ways is:

7! / (1! * 2! * 4!)

= (7*6*5*4*3*2*1) / (1 * 2*1 * 4*3*2*1)

= 105

However, since the objects are not in a line but in a circlular way (e.g. on a bracelet), the circle can be rotated to that what appears to be 7 possibilities is actually just 1, i.e. ABCDEFG = BCDEFGA = CDEFGAB = DEFGABC = EFGABCD = FGABCDE = GABCDEF. So instead of 105 possibilities, there are actually 105 divide 7 = 15 ways.

But since you can also FLIP the bracelet, not just rotate it, you'll find that of those 15 possibilities, 3 of them are symmetrical (so flipping them has no effect), and the other 12 are actually just 6 pairs of equivalent possibilities. So the actual answer is 15 - 6 = 3 + (12/2) = 3 + 6 = 9

## Bookmarks