Bunuel wrote:

A certain jewelry store sells customized rings in which three gemstones selected by the customer are set in a straight row along the band of the ring. If exactly 5 different types of gemstones are available, and if at least two of the gemstones in any given ring must be different, how many different rings are possible?

(A) 20

(B) 60

(C) 90

(D) 120

(E) 210

Case 1: All gemstones are different = 5C3 *3!/2 = 30

5C3 = Number of ways of choosing 3 out of 5 types of gemstones

3!/2 = 3! represents the arrangement of all gemstones but since CLOCKWISE and COUNTERCLOCKWISE arrangements are identical in case of circular arrangement in ring therefore we divide 3! by 2

Case 2: Two gemstones are same and one is different = (5C2*2)*[(3!/2!)/2] = 30

(5C2*2) = 5C2 is used to choose 2 types out of 5, Since one of the two chosen types has to be repeated so we choose the repeated gemstone in 2 ways

[(3!/2!)/2] Circular arrangement of three objects two of which are identical

Total Possible Rings = 30+30 = 60

Answer: option B

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