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# A certain jewelry store sells customized rings in which

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15 Oct 2004, 12:21
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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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15 Oct 2004, 13:18
Total ways: 5^3 = 125.
Unwanted: 5. (All the same)
125 - 5 = 120
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15 Oct 2004, 13:26
Thats correct ...
OA is D

When I went to work the problem I kept trying to do

5 possibilities for 1st stone
5 possibilities for 2nd stone
4 possibilities for 3rd stone

or 5*5*4 = 100.

I can see that this is not quite right (for one reason need to divide out identical cases), however can someone please explain why this method is wrong and, if possible, what it's trying to calculate?

Thanks
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15 Oct 2004, 16:24
tyagel, It is more complicated this way. You didn't count all the options.

There are 2 options:
1) All different = 5*4*3 = 60
+
2) 2 Equal, 1 Different = 5C2 * 2 * 3C2 = 60.

60 + 60 = 120.

Let me explain the second one:
5C2 : Pick 2 stones out of 5 ,
2 : Choose 1 of those two colors again.
3C2 : Pick two places for the identical colors, order doesn't matter.
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15 Oct 2004, 23:08
# all 3 different 5P3 = 5*4*3 = 60

# only 2 are same and 1 different (think of the two same as one) 4P2 = 12

There are 5 different stones that can be group together to form 2 same = 5 * 12

60 + (5*12) = 60+60 120 ways.
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16 Oct 2004, 06:17
Ok now I understand ...

Thanks for the help!

A slightly different way (although pretty much same a Dookie's) is

5P2*3C2

The first is a permutation since order can be considered to determine which color gets two.
16 Oct 2004, 06:17
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# A certain jewelry store sells customized rings in which

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