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A jewelry store sells customized rings

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by pnk » Sat Jul 24, 2010 11:10 am
A jewelry store sells customized rings in which 3 gems selected by the customer are set in a straight row along the band of the ring. If exactly 5 different gems are available and if at least 2 gems in any given ring must be different, how many different rings are possible?
20
60
90
120
210


OA - 120
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by kvcpk » Sat Jul 24, 2010 11:17 am
pnk wrote:A jewelry store sells customized rings in which 3 gems selected by the customer are set in a straight row along the band of the ring. If exactly 5 different gems are available and if at least 2 gems in any given ring must be different, how many different rings are possible?
20
60
90
120
210


OA - 120
Number of rings when all 3 gems are different:
5*4*3=60

Number of rings when 2 gems are same and the other is different:
Number of rings with the 2 same gems in first 2 positions 5*1*4 = 20
Now, the positions can be changed 3c2 = 3 ways.
Hence 20*3 = 60 rings are possible.

Total =60+60 = 120
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by GMATGuruNY » Sat Jul 24, 2010 11:34 am
pnk wrote:A jewelry store sells customized rings in which 3 gems selected by the customer are set in a straight row along the band of the ring. If exactly 5 different gems are available and if at least 2 gems in any given ring must be different, how many different rings are possible?
20
60
90
120
210


OA - 120
Remember this rule:

good = total - bad

total = 5 * 5 * 5 = 125 (number of ways to arrange 3 gems from 5 choices if gems can be reused)

bad = 5 (a bad arrangement is all 3 gems the same; since we have 5 gems from which to choose, there are 5 ways to have all 3 gems be the same)

good = total - bad = 125 - 5 = 120.

An efficient approach that requires less messy arithmetic.
Last edited by GMATGuruNY on Sat Jul 24, 2010 11:37 am, edited 2 times in total.
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by kvcpk » Sat Jul 24, 2010 11:36 am
GMATGuruNY wrote:
pnk wrote:A jewelry store sells customized rings in which 3 gems selected by the customer are set in a straight row along the band of the ring. If exactly 5 different gems are available and if at least 2 gems in any given ring must be different, how many different rings are possible?
20
60
90
120
210


OA - 120
Remember this rule:

good = total - bad

total = 5 * 5 * 5 = 125 (number of ways to arrange 3 gems from the 5 choices if gems can be reused)

bad = 5 (a bad arrangement is all gems the same; since we have 5 gems from which to choose, there are 5 ways to have all 3 gems be the same)

good = total - bad = 125 - 5 = 120.

An efficient approach that requires less messy arithmetic.
Thats Awesome!!
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by bendaniel » Sun Jul 25, 2010 7:47 am
Hi,
I'm not able to understand the concept of "total"....
how it is 5*5*5?... could somebody please explain this?... i, on the other hand assumed that to be 5!, which is 120....
Enlighten me please...
thank u.....
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by GMATGuruNY » Sun Jul 25, 2010 7:56 am
bendaniel wrote:Hi,
I'm not able to understand the concept of "total"....
how it is 5*5*5?... could somebody please explain this?... i, on the other hand assumed that to be 5!, which is 120....
Enlighten me please...
thank u.....
In the problem above, we are arranging 3 gems from 5 choices.

Total = total number of ways to arrange 3 gems from 5 choices if we can reuse gems. We have 5 choices for the 1st position, 5 choices for the 2nd position, 5 choices for the 3rd position: 5*5*5=125.

Bad = an arrangement in which all 3 gems are the same (because a good arrangement is one in which at least 2 gems are different). Since we have 5 gems to choose from, there are 5 ways to have the 3 gems all be the same. So bad = 5.

Good = Total - Bad = 125 - 5 = 120.

Does this help?
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by bendaniel » Sun Jul 25, 2010 8:07 am
s it helps... thanq guru.... :)
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by vishaljainsxc » Mon Sep 19, 2011 9:12 pm
Can you make me understand how


Bad = an arrangement in which all 3 gems are the same (because a good arrangement is one in which at least 2 gems are different). Since we have 5 gems to choose from, there are 5 ways to have the 3 gems all be the same. So bad = 5.
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by mankey » Wed Sep 21, 2011 5:15 am
What is wrong with the following approach?

Selecting 2 different gems out of 5: 5C2 * 3!/2!=30
Selecting 3 different gems out of 5: 5C3 * 3!=60
Total=30+60=90

Please clarify.

Thanks
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by GMATGuruNY » Wed Sep 21, 2011 6:45 am
vishaljainsxc wrote:Can you make me understand how


Bad = an arrangement in which all 3 gems are the same (because a good arrangement is one in which at least 2 gems are different). Since we have 5 gems to choose from, there are 5 ways to have the 3 gems all be the same. So bad = 5.
A bad arrangement is one in which all 3 gems are the same.
Number of options for the first gem = 5. (Any of the 5 gems can be chosen.)
Number of options for the second gem = 1. (Must be the same as the first gem.)
Number of options for the third gem = 1. (Must be the same as the first gem.)
To combine the options above, we multiply:
Bad arrangements = 5*1*1 = 5.
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by beatthegmat.garry » Wed Sep 21, 2011 7:09 pm
Why do you consider permutations here?
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