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Unformatted text preview: 1 Combinations & Permutations How many different ways can you chose K objects from N? How many ways can you chose K objects from N? 2 Learning Objective 4 : Factorials Rules for factorials: n!=n*(n1)*(n2)2*1 1!=1 0!=1 For example, 4!=4*3*2*1=24 Combinations The number of ways of arranging k successes in a series of n observations (with constant probability p of success) is the number of possible combinations (unordered sequences). This can be calculated with the binomial coefficient : )! ( ! ! k n k n n k C k n = = The n _perm_ k uses the factorial notation ! . The factorial n! for any strictly positive whole number n is: n! = n ( n 1) ( n 2) 3 2 1 Where k = 0, 1, 2, ......., or n Permutations The number of ways of arranging k successes in a series of n observations (with constant probability p of success) is the number of possible permutations (ordered sequences). This can be calculated with the: )! ( ! k n n p k n = The binomial coefficient n _choose_ k uses the factorial notation ! . The factorial n! for any strictly positive whole number n is: n! = n ( n 1) ( n 2) 3 2 1 Where k = 0, 1, 2, ......., or n Difference If the order doesn't matter, it is a Combination If the order does matter it is a Permutation . 1. Permutations with Repetition n * n * n.Choosing n, r times, allowed to repeat 2. Permutations without RepetitionChoosing n, and not allowed to repeat (Factorial) How many ways can first and second place be awarded to 10 people? Difference If the order doesn't matter, it is a Combination If the order does matter it is a Permutation . For example, let us say balls 1, 2 and 3 were chosen. These are the possibilities: Order does matter Order doesn't matter 1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1 1 2 3 Difference If the order doesn't matter, it is a Combination (No repeats) If the order does matter it is a Permutation . (Repeats) Combination: Picking a team of 3 people from a group of 10. C(10,3) = 10!/(7! * 3!) = 10 * 9 * 8 / (3 * 2 * 1) = 120. Permutation: Picking a President, VP and Waterboy from a group of 10. P(10,3) = 10!/7! = 10 * 9 * 8 = 720. Combination: Choosing 3 desserts from a menu of 10. C(10,3) = 120. Permut: Listing your 3 favorite desserts, in order, from a menu of 10. P(10,3) = 720. Difference If the order doesn't matter, it is a Combination (No repeats) If the order does matter it is a Permutation . (Repeats) Suppose we want to find the number of ways to arrange the three letters in the word CAT in different twoletter groups where CA is different from AC and there are no repeated letters. Because order matters, we're finding the number of permutations of size 2 that can be taken from a set of size 3. This is often written 3_P_2. We can list them as: CA CT AC AT TC TA When we want to find the number of combinations of size 2 without repeated letters that can be made from the three letters in the word CAT,...
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 Spring '11
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