Calculating Probability: Sample of Workers & Shifts | Probability Question

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To determine the probability that at least one shift is unrepresented when drawing 6 slips from a total of 45 workers (20 day, 15 swing, 10 graveyard), the approach involves calculating the complement of the probability that all shifts are represented. The total number of ways to select 6 workers from 45 is calculated, followed by determining the number of ways to select 6 workers that include at least one from each shift. This requires combinatorial calculations to find the valid combinations for each scenario. The final probability can be derived by subtracting the probability of all shifts being represented from 1. This problem emphasizes understanding probability concepts and combinatorial methods in a practical context.
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If a production facility employs 20 workers on the day shift, 15 workers on the swing shift, and 10 workers on the graveyard shift.

d) What is the probabily that drawing 6 slips (without replacement) that at least one of the shifts will be unrepresented in the sample of workers.

This is part D of a multi-part question from Probability class. I'm not sure how to solve it, but I'm guessing you can use the equation "1-P(that all shifts is represented)"... but how do you calculate the probability of that? Thank you
 
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This is a thinking question, not a calculating question. Even this may be too much of a hint.
 

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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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