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A certain jewelry store sells customized rings in which three gemston

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A certain jewelry store sells customized rings in which three gemston [#permalink]

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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

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A certain jewelry store sells customized rings in which three gemston [#permalink]

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New post 03 Feb 2018, 05:14
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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Re: A certain jewelry store sells customized rings in which three gemston [#permalink]

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total number of arrangements is 5^3 ( no restrictions). Each slot can have any of the 5 gems.
at least 2 different stones: the opposite of which is all stones stones are equal. At least 2 different stones is either ABB or ABC. Therefore the only arrangement for all stones are equal is AAA. Since we have 5 different gems therefore we have only 5.
The answer becomes 5^3-5=120.
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Re: A certain jewelry store sells customized rings in which three gemston [#permalink]

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New post 23 Mar 2018, 03:38
Gemstones can be selected in 5c3 ways and then arranged in 3! ways. Thus, total number of possible rings = (5c3) * 3! = 60

Thus, option B.

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Re: A certain jewelry store sells customized rings in which three gemston [#permalink]

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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


We use the formula:

#(rings with any choice of 3 gemstones) = #(rings where at least two gemstones are different) + #(rings where all the gemstones are the same)

Since there are 5 choices of gemstones and since a gemstone can be used more than once, the number of rings without any restrictions is 5^3 = 125.

The number of rings where all the gemstones are the same is equal to the total number of available choices for gemstones, which is 5. In other words, we are choosing one gemstone from 5 available choices and there are 5C1 = 5 ways to do this.

Thus, the number of rings where at least two gemstones are different is 125 - 5 = 120.

Answer: D
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Re: A certain jewelry store sells customized rings in which three gemston [#permalink]

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New post 27 Mar 2018, 09:56
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Answer is option D.

1.The question states that at least 2 is a must , so we can pick either 2 gems or 3 gems ( since 3 is the max that can be selected according to first line ).
This gives us 5C2+5C3 = 20.
2.The way the three rings can be arranged is given by 3!.

Multiplying 1 & 2 we get 20*3!=120.
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Re: A certain jewelry store sells customized rings in which three gemston [#permalink]

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2 answer options in the picture ?
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Re: A certain jewelry store sells customized rings in which three gemston   [#permalink] 09 Apr 2018, 11:18
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