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55010Note2 - J Chen Handout 2 STAT550 2010 1 STAT550...

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J. Chen, Handout 2, STAT550, 2010 1 STAT550 Applied Probability http://www-rohan.sdsu.edu/ jchenyp/STAT5502010.htm 1.3. Counting Techniques The permutations and combinations are very useful tools to compute the proba- bility of the event. In particular, both methods provide efficient ways to calculate the number of the sample points in sample space for the equally likely classical probability model. The terminology of the permutations and combination can be described as: A PERMUTATION is an arrangement of distinct items (objects) in a particular order A COMBINATION is a selection of items (objects) without considering order 1). Multiplication rule If procedure (operation) A can be done in m different ways and procedure B in n different ways, the sequence (A,B) can be done in m × n different way. In general, if procedure A i , i = 1 , · · · , k , can be performed in n i ways, i = 1 , · · · , k, respectively, then the order sequence ( A 1 , A 2 , · · · , A k ) can be performed in n 1 ×· · ·× n k different way. EX 1. A traveler may drive any one of three routes from Kansas City to Chicago and any one of four routes from Chicago to New York City as depicted in Figure. Solution: If we count the number of routes from Kansas City to Ne York through Chicago, we see there are a total of (3)(4) = 12 routes. EX 2. The combination lock on a briefcase has two dials, each marked off with 16 notches. To open the case, a person first turns the left dial in a certain direction for two revolutions and then stops on a particular mark. The right dial is set in a similar fashion, after having been turned in a certain direction for two revolutions. How many different setting are possible? Solution: By using the multiplication rule, opening the briefcase corresponds to the four-step sequence ( A 1 , A 2 , A 3 , A 4 ), then Number of different settings = n 1 .n 2 .n 3 .n 4 = 2 × 16 × 2 × 16 = 1024. 2). Permutations
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J. Chen, Handout 2, STAT550, 2010 2 A PERMUTATION is an arrangement of distinct objects in a particular order. Order is important. Suppose that we have n distinct items (objects) and we want to permute (or order) these items. Thinking of k positions, we will put one item in each position. These are n different ways to choose the item for position 1, n -
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