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singhpavi

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  1. Now that I made you guys curious. Here is the answer !! But i have no idea what the explaination is all about. The correct answer is 120. This question can be answered either by calculating the number of permissible rings directly or through a shortcut. We'll start with the direct way first. If at least 2 of the stones must be different, that means that the only permissible rings contain 3 different types of stone or 2 different types of stone. If we start with the rings that have 3 different types of stone, we have to choose 3 types of stone out of the 5 available. So that's http://jasper.kaptest.com/content/media/86/48186.1.apcq04e01.gif. So there are 10 different groups of 3 stones that can be chosen. But each group of 3 can be arranged in 3! or 6 different ways. That makes 10 x 6 or 60 different ways to select and arrange 3 different stones in a straight line. Now for the rings with 2 different types of stone: We're basically matching two types of stone, with 2 of one type and 1 of another. Each type of stone can be matched with the 4 other types of stone, so the 5 types of stone can each be matched 4 times. That's 5 x 4 = 20 combinations of 2 stones that are the same and 1 stone that's different. So we have 20 groups of 3 stones where 2 are the same and 1 is different. Each of these 20 groups can be arranged http://jasper.kaptest.com/content/media/86/48186.1.apcq04e03.gif in ways. This yields 20 x 3 = 60 ways to choose 2 of one type and 1 of another and then arrange them in a straight line. So altogether we have 60 rings where all three stones are different and 60 where 2 are the same and 1 is different, for a total of 120 possible rings. The shortcut is to realize that the only impermissible rings are those where all the stones are the same type. So if we were to figure out how many rings were possible overall and then subtract the ones that violate the "rules," we'd be left with the number of permissible rings. If we didn't have to worry about selecting different types of stone, there would be 5 x 5 x 5 = 125 possible rings, since for each place in the ring, there would be 5 types of stone available. Now, we have to subtract the number of rings where all the stones are the same type. If we have 5 types of stone, there are only 5 possible rings where all the stones are of the same type. So if we subtract those 5 from the 125 that are possible overall, we're left with 120.
  2. 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 gemstone are available, and if at least two of the gemstones in any given ring must be different, how many different rings are possible?
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