Understanding Sample Space: Explaining a Tricky Dice Question

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The discussion centers on the sample space of a six-sided die with the numbers 1, 1, 1, 2, 3, and 4. Participants debate whether the sample space should be represented as 1, 2, 3, 4 or 1, 1, 1, 2, 3, 4. It is clarified that the sample space consists of all possible outcomes without repetitions. Therefore, the correct sample space is 1, 2, 3, 4. This conclusion resolves the argument among classmates.
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A six sided dice has the numbers 1,1,1,2,3,4.
What is the sample space?
Is it 1,2,3,4 or 1,1,1,2,3,4?
Could you please explain why?
All of my classmates are arguing about this question.
 
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The "sample space" is the set of all possible outcomes. A set does not have repetitions.
 
Thnx. Now i know its 1,2,3,4.
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks

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