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# 2.06 - Section 2.6 Elementary Combinatorics Elementary...

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Section 2.6, Elementary Combinatorics 01/12/2010 1 2 Addition Rule for disjoint subsets of a finite set. If { A 1 , A 2 , ... , A n } is a collection of (pairwise) disjoint subsets of a finite set S , then Elementary Combinatorics – Counting Tools Definition. Suppose that A is a finite set. The cardinality of A is the number of elements in A , and is denoted | A |. Addition Rule for two subsets of a finite set. If A and B are subsets of a finite set S , then . 1 1 = = = n k k n k k A A U . B A B A B A + = 3 Multiplication Rule. Suppose a procedure consists of m steps, performed sequentially, and that: the 1 st step can be performed in n 1 ways the 2 nd can be done in n 2 ways regardless of how the 1 st was done : the m th can be done in n m ways regardless of the previous choices. Then the number of different ways to perform the entire procedure is n 1 n 2 ··· n m . Equivalently, the number of ways to fill m blanks, where: the 1 st blank can be filled in n 1 ways the 2 nd can be filled in n 2 ways regardless of how the 1 st was filled etc. is n 1 n 2 ··· n m. Equivalently, the number of m -tuples where the 1 st coordinate can be chosen in . . . . 4 . 1 1 = = = n k k n k k A A U . B A B A B A + = Multiplication Rule: A procedure consists of m steps, performed sequentially, and for each k , step k can be performed in n k ways regardless of the choices made on the previous steps. Then the number of different ways to perform the entire procedure is n 1 n 2 ··· n m . Addition Rule for two subsets of a finite set, S : Addition Rule for disjoint subsets of a finite set, S : Examples. For each experiment, determine the number of possible outcomes. Equivalently, determine the cardinality of the (natural or obvious) sample space for each. 1. Toss either a coin or a die, but not both. | S | = 2. Toss both a coin and a die. | S | = 3. Toss a die three times. | S | = 5 . 1 1 = = = n k k n k k A A U . B A B A B A + = Multiplication Rule: A procedure consists of m steps, performed sequentially, and for each k , step k can be performed in n k ways regardless of the choices made on the previous steps. Then the number of different ways to perform the entire procedure is n 1 n 2 ··· n m . Addition Rule for two subsets of a finite set, S : Addition Rule for disjoint subsets of a finite set, S : Examples. For each experiment, determine the number of possible outcomes. Equivalently, determine the cardinality of the (natural or obvious) sample space for each. 4. Choose one card from a pile consisting of all the hearts and face cards from a standard deck of cards. | S | = 5. Arrange three people in a line (1 st , 2 nd , 3 rd ). | S | =

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