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Lesson 9 - Lesson 9 Combinatorix Here we begin methods of...

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Lesson 9: Combinatorix Here we begin methods of counting First we need factorial notation: n ! = n ( n - 1)( n - 2) · · · 2 · 1 Examples: 0! = 1, 1! = 1, 3! = 3 * 2 * 1 = 6, 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720 Counting with order Lets say we have 5 books. How many ways can we put them all on the bookshelf? In the first spot, there are 5 books we can pick. In the second spot there are 4 books left. In the third spot, there are 3 possibilities. In the fourth spot, there are 2 books to choose from. Finally, there is only one book left in the last spot. So we get 5! ways to order the books. So for n books, there are n ! ways to order them without repetition. If we have m books and n spots on the bookshelf, we get m ! / ( m - n )! ways to order them without repetition or m n ways to order them with repetition. Try it for 5 books and 3 spots. Counting without order What if we don’t care about order. Lets say we have 5 books. How many ways can we pick 3 to bring to school? Two ways to get this. There are 5! / (5 - 3)! ways to get an ordered list of 3 books and there are 3! ways to order any three books. Since every possible
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