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Learning Objectives
* Determine whether a situation involves permutations or combinations
* Understand the mathematical implications of the words 'and' & 'or'
Having covered the basics of combinations and permutations, you are ready to have a mixture of problems with slight variations. A common variation involves an understanding of some key words used in mathematics. Commonly, the word "and" indicates multiplication and the word "or" indicates addition. Consider the examples below.
Example 1
In how many ways can a committee of 3 people be chosen if there are 8 men and 4 women available for selection and we require that two men and one woman be on the committee?
Solution
The order that we place the people on a committee does not matter. It makes no difference if you are the first person or the last person selected for the committee. Either you are on the committee or you are not on the committee, therefore this is a combination question. Notice that we want two men and one woman. The word 'and' indicates multiplication. In other words, we will look for the product of how many ways we can select two men from eight and one women from four.
There are 112 ways to select this committee of 3 people.
Example 2
In how many ways can a committee of 5 people be chosen if there are 7 men and 5 women available for selection and we require at least 4 women on the committee?
Solution
We first ask "Does order matter?". In this case, the order that someone is placed on a committee does not matter. Either you are on the committee or you are not. Once again, we are dealing with a combination question. The key phrase in this example is at least. This can be interpreted to mean that we either select 4 women and 1 man or 5 women and 0 men.
Remember that the word 'and' indicates multiplication and the word 'or' indicates addition. It looks like we are going to have some addition and some multiplication in this problem. The words "at least" or "at most" will indicate an "or" situation.
There are 36 ways to put this committee together.
Example 3
In a certain country, there are two political parties. Each party is responsible for nominating both a presidential and vice-presidential candidate. The candidates will participate in a debate once they are chosen. In the first party, there are 6 candidates available and in the second party there are 5 candidates available. How many different debate combinations are possible?
[Figure2]
Solution
The order that we select the candidates does make a difference. Selecting one person (person A) for a presidential candidate and another person (person B) for a vice-presidential candidate is different than selecting person B for presidential candidate and person A for a vice-presidential candidate. Therefore, this is a permutations question. Since we will select candidates from the first party and candidates from the second party, we expect there to be multiplication in this problem as well.
There are 600 different ways that the debate participants can be chosen.
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