Discussion1

Discussion1 - Discussion 1 Emily Mower 1 Combinations and...

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Discussion 1 Emily Mower January 21, 2010 1 Combinations and Permutations Combinations - order doesn’t matter Permutations - order matters Short example: The fruit salad is a combination of kiwis, strawberries, and blueberries. The lock is opened with the code 727 ( permutation ) 1.1 Permutations There are two types of permutations: No repetition (e.g., there are two open offices, president and vice- president) Repetition (e.g., the key to the lock is 727) 1
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1. Permutations without Repetition In this case, the goal is to choose r items out of n items. An item can be selected only once. Thus, at each iteration there is one fewer item to choose from: n * ( n - 1) * ( n - 2) * ... * ( n - r + 1) (1) Example: Q. How many ways are there to select 16 numbered pool balls (num- bered from 1 - 16)? A. 16 * 14 * 13 * ... * 2 * (16 - 16 + 1) = 16 * 14 * 13 * ... * 2 * 1 = 20 , 922 , 789 , 888 , 000 Example: Q. How many ways are there to select 3 balls out of the 16 numbered balls? A. 16 * 15 * ... * (16 - 3 + 1) = 16 * 15 * 14 = 3 , 360 This can be expressed mathematically: Number of ways to order n items: n * ( n - 1) * ... * 2 * 1 = n ! Number of ways to order r out of n items: n ! ( n - r )! Summary of permutations without repetition: There are n items Choose r of them Total number: n ! ( n - r )! 2
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2. Permutations with Repetition These are the easiest to solve. The goal is to choose r items out of n items. An item can be selected repeatedly. Thus, at each iteration n items can be selected: r Y m =1 n = n * n * ... * n = n r (2) Example: Q. How many ways are there to choose three numbers from the set 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A. 10 * 10 * 10 = 10 3 = 1000 Summary of permutations with repetition: There are n items Choose r of them Total number: n r 1.2 Combinations There are two types of combinations: No repetition (e.g., lottery numbers: 5,6,7,19 ) Repetition (e.g., coins in a pocket: 5,5,5,10,25) 3
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1. Combinations without Repetition In this case, the goal is to choose r items out of n items where the order does not matter. An item can be selected only once. Thus, at each iteration there is one fewer item to choose from. First calculate the permutation ( r items out of n items): n ! ( n - r )! Then alter the result, removing the effect of order: n !
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