Arrangement problem

Question

The number of distinct bracelets of five beads made up of red, blue, and green beads (two bracelets are indistinguishable if the rotation of one yield another) is,
a. 243
b. 81
c. 51
d. 47

Answer 1

Lets Solve this by patterns possible and there are

k!/(k−n)! ways to put k things into n ordered slots.

here k =3 and n ordered slots can understood as:

 

RRRRR here all beads of red color so it is 1 ordered slot

RRBBR here there are 2 things used so it is 2 ordered slot.

Now Let R =x, G=y, B=z

possible combinations

xxxxx: 3 = 3!/(3−1)!

xxxxy: 6 = 3!/(3−2)!

xxxyy: 6 = 3!/(3−2)!

xxxyz: 6 = 3!/(3−3)

xxyyz: 6 = 3!/(3−3)!​​​​​​​

xxyxy: 6 = 3!/(3−2)!

xxyxz: 6 = 3!/(3−3)!

xxyzy: 6 = 3!/(3−3)!

xyzyz: 6 = 3!/(3−2)!

3 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 = 51