Lecture_4 - 1 Combinations & Permutations How...

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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*(n-1)*(n-2)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 Repetition-Choosing 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 two-letter 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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Lecture_4 - 1 Combinations & Permutations How...

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