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COMBINATORICS Notes

# COMBINATORICS Notes - COMBINATORICS 1 Example A senator...

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COMBINATORICS 1. Example: A senator must hire a new chief-of-staff and a new press secretary. If there are 5 applicants for chief-of-staff and 3 applicants for press secretary, how many different ways can these two positions be filled? Solution: Let’s call the applicants for chief-of-staff A, B, C, D, and E. We’ll name the applicants for press secretary X, Y, and Z. The following tree diagram shows all the possibilities: We see that there are __ 15 ___ different ways to make the two hires. Do you see a shortcut we could have used to get the answer without drawing the tree diagram. diagram? 5 X 3 = 15 This can be generalized to situations where more than two consecutive choices are being made. 2. Example: All of a sudden, the senator’s director of legislative affairs quits. If there are two applicants for this position, in how many ways can the senator fill the three positions? Solution: Using the Fundamental Principle of Counting, 5 X 3 X 2 = 30 3. Example: A voter has to rank 5 alternatives. How many different preference lists are possible? Solution: 5 X 4 X3 X 2 X 1 = 5! = 120 Page 1 of 4 Fundamental Principle of Counting: If the first choice can be made in x ways and the second choice can be made in y ways, then the choices can be made consecutively in xy ways.

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COMBINATORICS Notes - COMBINATORICS 1 Example A senator...

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