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If $n$ identical dice are rolled, then the number of possible outcomes are?

My Approach:

Here, since the number of outcomes are asked, the answer should be $6^n$ since there are 6 options for a dice and it is being rolled $n$ times but I don't understand how we are supposed to use stars and bars here to get the correct answer as ${}_{n+5} \mathrm{ C }_5$

Patrick Schick
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    This can't be answered unless you specify which outcomes are meant to be equivalent. $6^n$ is clearly correct if the dice are considered distinct. If not, then presumably you just look at $6-$tuples which count the number of $1's,2's,\cdots$ that you get. – lulu Aug 31 '21 at 13:13
  • If the dices are identical, I would rather expect that the order plays no role. But in an exercise, such things have to be clearly stated, othewise it is ill-posed. – Peter Aug 31 '21 at 13:19
  • Suggested reading: Why is flipping a head then a tail considered a different outcome than a tail then a head?. The punchline is that the sample space in probability can contain as much or as little information as you like so long as it can be used to describe the events you are interested in and that which sample space you use is a choice. Certain choices are more useful than others, for instance by being equiprobable. – JMoravitz Aug 31 '21 at 13:50
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    That said, the way this problem is phrased, it sounds like they are wanting a count of outcomes where order of outcomes doesn't matter in which case the stars-and-bars approach is indeed what they wanted. It is worth emphasizing however that these outcomes counted by stars and bars are not equally likely and will not be useful for probability questions about this scenario, the $6^n$ outcomes where order mattered being far more useful in the long run. – JMoravitz Aug 31 '21 at 13:51

1 Answers1

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In general , we think that dice are different , so there are $6^n$ different possible outcomes when dice are distinct .

However, when the dice are identical , we only consider that how many different number will appear ,we can do it by stars and bars such that

Assume that the gaps between the bars means numbers on die . For example $$1|2|3|4|5 |6 $$ .Now , if we have $n$ stars , how many ways are there to see possible out comes ?

Answer is $$C(n+6-1, 5)$$ . Because ,we think the arrangements of $n$ identical stars and $5$ identical bars. You can think it like combination with repetition.

For example , if $n=4$ then $*|**|||*|$ means $(1,2,2,5)$ , $|||||****$ means $(6,6,6,6)$

If $n=3$ ,then $||*||*|*$ means $(3,5,6)$

Not a Salmon Fish
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