What are Factorials, Permutations, and Combinations?

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Factorials, permutations, and combinations are fundamental concepts in combinatorics. The discussion revolves around evaluating specific mathematical expressions like P(10,3) and 9!. The user expresses confusion regarding a particular expression, questioning if it represents a combination, specifically 25C5. Clarification is provided that the user is dealing with a list of questions, some involving factorials and others permutations. Understanding these concepts is essential for solving problems in combinatorial mathematics.
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Homework Statement



I am asked "evaluate the following factorials, permutations, and combinations". I am given a series of questions I understand, like P(10,3), 9!, etc. But then I have a series like this:

9a3b4bfb4a4f.jpg


Homework Equations




9a3b4bfb4a4f.jpg



The Attempt at a Solution



I am not sure what this is, is it a combination, perhaps because there hasn't been any yet, and this is solved by 25C5?
 
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I'm not sure what you mean by a series, but usually , yes, it is 25C5.
 
Bacle2 said:
I'm not sure what you mean by a series, but usually , yes, it is 25C5.

Sorry, I ment a list, as in I have 10 questions, with a few looking like the one on the photo, a few looking like 9!, and a few looking like p(10,3). =)
 
Oh, O.K, makes sense, maybe I was kind of thick.
 

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 'Greatest possible value of a constant in polynomial'

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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