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How many positive integers of two different digits

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How many positive integers of two different digits can be formed from the integers 1,2,3, and 4? 2. How many different arrangements of six distinct books each can be made on a shelf with space for six books?
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COUNTING Multiplication Principle If an activity can be constructed in t successive steps and step1 can be done in n 1 ways, step 2 can be done in n 2 ways, … , and step t can be done in n t ways, then the number of different possible activities is n 1 ∙n 2 ∙∙∙n t
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COUNTING Example Given: 1. List all possible dinners consisting of one main course and one beverage. 2. List all possible dinners consisting of one appetizer, one main course and one beverage. APPETIZER S MAIN COURSES BEVERAGE S Nachos Hamburger Tea Salad Cheeseburger Milk Fish Fillet Soda Root Beer
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COUNTING Examples 1. How many different arrangements of each consisting of four different letters can be formed from the letters of the word “PERSONAL” if each arrangement is to begin and end with a vowel. 2. Suppose that three of six books are math books and three are art books. a. How many different arrangements of the six books can be made on the shelf
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COUNTING b. How many different arrangements of the six books can be made on the shelf if math books are to be kept together but three art books can be placed anywhere? 3. In how many ways can we select two books from different subjects among five distinct computer science books, three distinct math books and two
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COUNTING Definition (Permutation) A permutation of n distinct elements x 1 , …,x n is an ordering of the n elements x 1 , …, x n .
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COUNTING Definition (Permutation) Let S be a set containing n elements and suppose r is a positive integer such that r≤n . Then a permutation of r elements of S is an arrangement in a definite order, without repetitions, of r elements of S . notations: n P r ; P(n,r) ; P n,r ; “permutations of n taken r
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COUNTING Theorem The number of permutations n of elements taken r at a time is given by either of the following formulas: a. n P r = n(n-1)(n-2)…(n-r+1) b. n P r = n!/(n-r)!
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COUNTING Example 1. A bus has seven vacant seats. If three additional passengers enter the bus, in how many ways can they be seated? 2. In how many ways can four boys and four girls be seated in a row containing eight seats if a) a person may sit in any seat b) boys and girls must alternate seats
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COUNTING Theorem If we are given n element, of which exactly m 1 are alike of one kind, exactly m 2 are alike of a second kind, …, and exactly m k are alike of the kth kind, and if n = m 1 + m 2 +…+ m k then the number of distinguishable permutations that can be made of the n elements taking them all at one time is . ! m !... m ! m ! n k 2 1
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COUNTING Example 1. How many strings can be formed using the following letters MISSISSIPPI?
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