Notes M362K Chp1 - M362K Dr.Berg Chapter 1.Combinatorics...

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M362K Dr.Berg Chapter 1…Combinatorics 1.1 The Basic Counting Principle We begin with the counting techniques often used to calculate probabilities. Theorem If two tasks ( experiments, choices, etc. ) can be done independently in m different ways for the first and n different ways for the second, then there are mn ways to perform the tasks. Proof: Enumerate the outcomes of the first task 1 through m , and enumerate the outcomes of the second task 1 through n . Then the combined outcomes can be enumerated using 2–tuples ( i , j ) forming an m by n array containing mn distinct elements. Note: Mathematical induction generalizes this to any finite number of tasks. Example A (2b page 3) A college planning committee consist of 3 freshmen, 4 sophomores, 5 juniors, and 2 seniors. A subcommittee of 4, consisting of 1 person from each class, is to be chosen. How many different subcommittees are possible? Solution : We assume that the choices are independent, so there would be 3 4 5 2 =120 possibilities. Example B You have 3 pairs of shoes, 3 different colored slacks, and 5 different shirts. You must choose one of each type for an outfit. If we assume independence, how many different outfits can be chosen? Solution : There would be 3 3 5 = 45 different outfits. 1.2 Permutations The different orderings (left to right) of a finite set are called permutations of that set. These are also called arrangements, rearrangements, orderings, etc. Proposition There are n != n ( n - 1)( n - 2) 3 2 1 permutations of n objects. More generally, there are n ( n - 1) ( n - r +1) = n ! ( n - r )! permutations of r objects chosen from among n objects. page1
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M362K Dr.Berg Proof: There are n choices for the first object, then n –1 choices for the second, etc. Example A (3a page 4) How many different batting orders are possible for a baseball team consisting of 9 players?
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