hey_thr67 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?
A: 20
B: 60
C: 90
D: 120
E: 210
OA is D
Doing the basics wrong.
I have the answer as 5P3.
What does it mean by atleast 2 gems.
Good rings = total possible rings - bad rings.
Total possible rings:
For each of the 3 positions in the ring, any of the 5 gems could be selected.
Number of options for the first position = 5.
Number of options for the second position = 5.
Number of options for the third position = 5.
To combine these options, we multiply:
5*5*5 = 125.
Bad rings:
In a good ring, at least 2 of the gems are different.
Thus, in a bad ring, all 3 gems are the same.
Thus, if the gem types are A, B, C, D and E, there are only 5 bad rings:
AAA, BBB, CCC, DDD, EEE.
Good rings = 125-5 = 120.
The correct answer is
D.
At least 2 gems must be different means that all 3 gems cannot be the same.
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