Probability: # of Permutations from r Choices of n Objects

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The discussion centers on calculating the number of permutations when selecting r objects from n total objects, where p of those are identical. It explores different scenarios, including whether the p identical objects are the only identical ones or if there are more. The approach involves considering cases based on the number of distinct objects chosen alongside the identical ones. Clarification is sought regarding the arrangement of objects, particularly whether the identical objects are grouped or spread across different containers. The conversation highlights the complexity of permutations in combinatorial problems involving identical objects.
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There are n objects, within which there are p identical objects. Now, suppose if you randomly choose r objects from the n objects. How many permutations will you get from the r objects you choose?
 
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So we have n balls, p are identical. Are the rest not identical? I.e. are exactly p identical or at least p identical?

Suppose it's exactly p. One way is to do it by cases. Choose k from the n-p distinct objects and r-k from the p identical. In that case you have a sum over k=0...r.

Now imagine r identical balls, choose r-k to remain identical. Now paint the remaining k with distinct numbers from 1 to n-p. How many possibilities?
 
Michael Si said:
There are n objects, within which there are p identical objects. Now, suppose if you randomly choose r objects from the n objects. How many permutations will you get from the r objects you choose?
Your question is ambiguous. Are there n containers each with p objects? And the objects are identical only within each container- different containers contain different objects? Or are there a total of p (or np?) identical objects?
 

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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