NUmber of ways to answer a question?

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A multiple choice question with 5 answers allows for various combinations of selections, where each answer can either be chosen or not. This results in 2 options (choose or not choose) for each of the 5 answers. Therefore, the total number of ways to answer the question is calculated as 2 raised to the power of 5, equating to 32 different combinations. This includes the option of not selecting any answers at all. Understanding this concept can help in estimating probabilities in similar scenarios.
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A multiple choice question has 5 answers, anything from one to all the choices can be marked(more than one choice can be marked at a time). How many ways are there to answer the question? While waiting for on my chem question, I want to see how hard it would be to guess my way to the answer:smile: I think I did this in precalc last year..something about series..I can't really remember.
 
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HINT: You can either choose an answer or not choose an answer.
 

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

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
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Thread 'Making the denominator of a certain fraction real'

Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
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