mgf1107notes2 - COMBINATORICS Example: A senator must hire...

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COMBINATORICS 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? 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. This can be generalized to situations where more than two consecutive choices are being made. 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)(3)(2) = 30 ways. Example: A voter has to rank 5 alternatives. How many different preference lists are possible? Solution: (5)(4)(3)(2)(1) = 120 Products that start with a number and multiply by every integer down to one are very common in mathematics and we have a special notation for them. (5)(4)(3)(2)(1) = 5! (pronounced five factorial ) In general: n! = n(n-1)(n-2). ...(3)(2)(1) if n is a positive integer 0! = 1
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Example: Simplify: a) 7! b) ! 16 ! 18 c) ! 3 )! 3 12 ( ! 12 Solution: a) (7)(6)(5)(4)(3)(2)(1) = 5040 b) 306 17 18 ! 16 ! 16 17 18 = = c) 220 5 11 4 1 2 3 10 11 12 1 2 3 ! 9 ! 9 10 11 12 ! 3 ! 9 ! 12 = = = = A permutation is an arrangement of objects in a particular order.
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This note was uploaded on 04/29/2008 for the course MGF 1107 taught by Professor Storfer during the Spring '08 term at FIU.

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mgf1107notes2 - COMBINATORICS Example: A senator must hire...

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