Counting problem involving picking delegates

In summary, there are 6 officers and 94 non-officers in an organization of 100 members. The organization plans to elect 2 delegates to attend a convention, one of whom must be an officer and the other cannot be an officer. An alternate delegate, who can be either an officer or not, will also be elected in case one of the regular delegates is unable to attend. The total number of different outcomes for this election is 55272, taking into account that two people are already chosen as delegates.
  • #1
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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  • #2
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...
 
  • #3
Oh i see. The officer and the nonofficer that was chosen. makes sense
 

What is the counting problem involving picking delegates?

The counting problem involving picking delegates is a mathematical concept that deals with finding the number of ways in which a certain number of delegates can be chosen from a larger group of individuals. It is a fundamental concept in combinatorics, the branch of mathematics that studies counting and arrangements of objects.

What is the formula for solving this counting problem?

The formula for solving the counting problem involving picking delegates is nCr = n! / r!(n-r)!, where n is the total number of individuals and r is the number of delegates to be picked. This is also known as the combination formula.

What is the difference between permutations and combinations?

Permutations and combinations are both ways of counting and arranging objects, but they differ in their order. Permutations take into account the order in which objects are arranged, while combinations do not. In the context of picking delegates, permutations would count the number of ways in which specific individuals can be chosen for specific positions, while combinations would only count the total number of ways in which a certain number of delegates can be chosen.

How do we apply this concept in real-life scenarios?

The counting problem involving picking delegates can be applied in various real-life scenarios, such as elections, appointments, and team selections. For example, in an election, the number of ways in which a certain number of candidates can be chosen from a larger pool of individuals can be calculated using this concept.

What are some common mistakes to avoid when solving this type of problem?

Some common mistakes to avoid when solving the counting problem involving picking delegates include not considering the order of objects, not using the correct formula, and not accounting for repetitions. It is important to carefully read the problem and understand what is being asked before attempting to solve it.

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