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In how many ways can $5$ rings of different types be worn on $4$ fingers?


According to me,first finger have $5$ ways,second finger have $4$ ways, third finger have $3$ ways and last finger have $2$ ways.


Therefore there are $5 \cdot 4 \cdot 3 \cdot 2 = 120$ arrangements.

But in my textbook it's answer is $4^5$.

N. F. Taussig
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J.Doe
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    That is a bit strange: the answer of your book does not consider the order of the rings on each finger. – Robert Z Sep 09 '18 at 17:06
  • please answer this question math.stackexchange.com/questions/2910579/… since I am new on this site I don't know how to attract users towards my question.Please tell me how to gain attention of other users.And please answer this question – J.Doe Sep 09 '18 at 17:11

3 Answers3

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Your textbook goes for: "each of the $5$ rings has a choice out of $4$ fingers."

drhab
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    @RafaelNadal Then practicize the rule: no information=no limitation. But I admit that the question is not very well stated. You could also wonder: does the order of the rings on the fingers play a part? If e.g. all rings are on the same fingers then still there are $5!$ arrangements. – drhab Sep 09 '18 at 17:02
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    I do not like the answer in the textbook since at least two rings must be placed on the same finger and the order in which distinct rings are placed on a finger matters. – N. F. Taussig Sep 09 '18 at 17:05
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    @N.F.Taussig I fully agree with that. – drhab Sep 09 '18 at 17:06
  • thanks I got it I think @rafael is answer – J.Doe Sep 09 '18 at 17:08
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    please answer this question https://math.stackexchange.com/questions/2910579/how-to-identify-whether-permutation-question-involves-repetition-or-not since I am new on this site I don't know how to attract users towards my question.Please tell me how to gain attention of other users.And please answer this question – J.Doe Sep 09 '18 at 17:10
  • @J.Doe I know only one way to attract attention: a bonus. But you have no reputation enough for that (yet), I guess. – drhab Sep 09 '18 at 17:14
  • @drhab what is bonus?anyways I got your attention so please answer my question sier – J.Doe Sep 09 '18 at 17:16
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This question is not well stated. This question doesn't shows information about the limitation of number of ring on a particular finger. According to you it's limitation is 1 but in your book it doesn't have limitation.

Hercules
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Assuming the order of rings on each finger is not important, you can look at the problem in the following way: take the first ring, you have 4 options (fingers) for it, right? Next ring, you have the same number of options, so it is $4 \times 4 = 4^2$, and so on. For 5 rings it is obviously $4^5$.

Alex
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