Counting problem involving picking delegates

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SUMMARY

The problem involves selecting delegates from an organization of 100 members, including 6 officers. The election requires one delegate to be an officer and the other to be a non-officer, with an additional alternate delegate who can be either. The initial calculation of 55836 outcomes is incorrect; the correct total is 55272 outcomes, as the selection of the alternate must account for the already chosen delegates. This discrepancy arises from not properly adjusting the pool of candidates after the first two selections.

PREREQUISITES
  • Understanding of combinatorial selection principles
  • Familiarity with basic probability concepts
  • Knowledge of permutations and combinations
  • Ability to perform arithmetic operations with large numbers
NEXT STEPS
  • Study combinatorial mathematics, focusing on delegate selection problems
  • Learn about permutations and combinations in detail
  • Explore examples of similar election problems in combinatorial contexts
  • Practice solving problems involving conditional selections and adjustments
USEFUL FOR

Students studying combinatorial mathematics, educators teaching probability concepts, and anyone interested in understanding election delegate selection processes.

torquerotates
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Homework Statement

An organization of 100 members, 6 of whom are officers, plans to elect delegates to attend a convention. There are to be 2 delegates; one must be an officer and the other cannot be an officer. In addition, an alternate delegate, either an officer or not, will be elected and will attend if one of the regular delegates is unable to do so. How many different outcomes can this election have?

Homework Equations


The Attempt at a Solution

So, there are 6 officers to choose from and 94 non officers to choose from. If one of them are unable to be a delegate then there are 99 people to choose from(100-1 for the other person that cannot make it). So in all there should be 6*94*99=55836 different ways of picking delegates. But the back of the book gave 55272 ways. Is there something I did that was wrong?
 
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torquerotates said:

Homework Statement

An organization of 100 members, 6 of whom are officers, plans to elect delegates to attend a convention. There are to be 2 delegates; one must be an officer and the other cannot be an officer. In addition, an alternate delegate, either an officer or not, will be elected and will attend if one of the regular delegates is unable to do so. How many different outcomes can this election have?



Homework Equations





The Attempt at a Solution

So, there are 6 officers to choose from and 94 non officers to choose from. If one of them are unable to be a delegate then there are 99 people to choose from(100-1 for the other person that cannot make it). So in all there should be 6*94*99=55836 different ways of picking delegates. But the back of the book gave 55272 ways. Is there something I did that was wrong?

Two people are already out of the pool when you are choosing the third delegate...
 
Oh i see. The officer and the nonofficer that was chosen. makes sense
 

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