Combinatorics Problem: Selection of Job Applicants

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SUMMARY

The combinatorial problem involves determining the rankings of eight job applicants by three judges. The total number of ways the judges can produce their rankings is calculated as C(8,3)^3, resulting in 175616 combinations. For the probability of Mr. Dickens being selected, the calculation adjusts to C(7,2)^3, yielding a probability of 27/512 or approximately 0.053. The discussion highlights the importance of correctly interpreting the problem statement to arrive at accurate solutions.

PREREQUISITES
  • Understanding of combinatorial mathematics, specifically combinations and permutations.
  • Familiarity with the formulas C(n,r) = n!/r!(n-r)! and P(n,r) = n!/(n-r)!
  • Basic probability theory, particularly in calculating probabilities from combinatorial outcomes.
  • Ability to interpret and analyze mathematical problems accurately.
NEXT STEPS
  • Study advanced combinatorial techniques, including the use of generating functions.
  • Explore probability distributions and their applications in real-world scenarios.
  • Learn about ranking algorithms and their relevance in decision-making processes.
  • Practice additional problems involving combinations and permutations to solidify understanding.
USEFUL FOR

Students studying combinatorics, educators teaching probability and statistics, and anyone involved in decision-making processes that require ranking or selection methodologies.

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Homework Statement



There are eight applicants for the job of dog catcher and three different judges who each rank the applicants.Applicants are chosen if and only if they appear in the top three in all three rankings

a) How many ways can the three judges produce their three rankings?

b) What is the probability of Mr. Dickens, one of the applicants, being chosen in a random set of three rankings?

Homework Equations



C(n,r) = n!/r!(n-r)! and P(n,r) = n!/(n-r)! Everything we are doing at this point involves these two formulas.

The Attempt at a Solution



For (a), I think the total number of ways the three judges can produce their three rankings is
C(8,3)^3 = 175616.

For (b), if I pick Mr. Dickens, then that leaves me each judge with seven people to choose from, thus the amended equation becomes C(7,2)^3 = 9261, and the probability is 9261/175616 = 27/512 = 0.053

I think this is right, but I am not sure and would appreciate a second set of eyes on this one. Thanks for any help.
 
Last edited:
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I just realized I didn't even read the problem correctly, and that I need to find a probability for (b). So I changed the result for (b). Sorry for any confusion.
 

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