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 diagramshows 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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