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Multiplication Principal: If you want to chose one element from a set A containing n elements and one element from a set B containing m elements and one element from a set C containing p elements and so on, Then the number of ways in which you can choose these elements are given by the formula: nC1 × mC1 × pC1 × ... =n × m × p × ... Example1 You and a group of friends love to go to the movies.You always purchase popcorn, soda, and candy. You can choose from buttered or unbuttered popcorn, five different kinds of soda, and 18 different kinds of candy. Determine the number of possible configurations.x x = Example2 You have three pairs of jeans, five t-shirts, and two pairs of shoes. Determine the number of possible ways to dress up.x x = Example3Eight people are boarding an aircraft. Threehave tickets for first class and board beforethose in the economy class. In how manyways can the eight people board the aircraft? × = r-lists Repetition is allowed and the order is important Formula: If a set contains n elements and you want to choose r elements each time then the number of ways in which this can be done is: n ×n×n×...×n = nr Direction: Drag each text to its correct place Repetition not allowed - order important Repetition not allowed - order not important Repetition Allowed - order important r-lists ? Permutation ? Combination ? Combination Permutation r-lists nr ? n! / (r! (n-r)!) ? n!/ (n-r)! ? Example1How many 2-digit numbers containing only the digits 3,4,5,6 are there? Examples of these numbers are 33, 45, 54, 55 ... Here we have the set {3,4,5,6} and we want to chose 2elements of it with repetition and the order is of courseimportant (since 34 ≠43 ).So we must use the r-lists: Here we have: Then, there are n= nr= and = r = The answer can be found using multiplication principal: The numbers are as follows: 33, 34, 35, 3643, 44, 45, 4653, 54, 55, 5663, 64, 65, 6616 numbers We must chose two digits to fill in the two boxes: The first box can be filled with 4 digits and the secondcan be filled with 4 digits also, so the is 4×4 = 16 Example2:Suppose we have a coin (one face is called head (H) and the other is called tail (T) ). We tossedthis coin three times in air and we noted its upper face in each time: such as THT which means we obtained Tail then Head then Tail respectively. Question: How many different outcomes we have? Solution we have to choose 3 elements from the letters {H, T}The order is important and we must have repetitions. So: Then nr= n= and r= We can think of this question in the following way: We have to fill the three boxes letters {H or T}: So we have 2 choices in the first box, 2 choices in the second box and 2 choices in the third box, Using multiplication principal: The number of outcomes is : 2×2×2 = 23=8 H T T H H T To find all the previous outcomes, we use a Tree-diagram: Conclusions: Ex:1 How many anagrams are there of the letter SEAT? (An anagram is a word or phrase formed by rearranging the letters of a different word, typically using all the original letters exactly once) Applications Ex:2 Eight horses are running a race. How many different ways can these horses come in first, second, and third? Ex:3 In a certain state, each automobile license platenumber consists of two english letters followed by afour-digit number. How many distinct license plate numbers can be formed? In a certain Country, each automobile license plate consists of three digits followed by three English Capital letters. How many distinct license plate numbers can be made with no repeat letters. Ex:4 × = Ex: 5 How many four-digit numbers can be formed? Ex: 6 How many four digit numbers can be formed with distinct digits? Ex: 7 How many four digit numbers can be formed less than 5000? Ex: 8 How many different ways can a committeeof three people be selected from a club of 40 people? Ex: 9 How many ways can six people sit in a six-passenger car? A minivan has 8 seats (including the driver seat). A group of 8 friends want to go to a trip in this minivan. How many possible ways they can sit in this minivan if there are only four drivers among them? Ex: 10 × = Ex: 11 There are 42 numbers in the Lebanese Lotto in which you have to choose 6 numbers in every ticket.How many possible distinct tickets are there? The CEO of a company interviews ten people forfour equal positions. Four of the applicants are women.If all ten are qualified, in how many ways can theCEO fill the four positions? Ex: 12 The CEO of a company interviews ten people forfour same positions. Four of the applicants are women.If all ten are qualified, in how many ways can theCEO fill the four positions if exactly two selectionsare women? Ex: 13 Ex: 14 In how many ways can a committee (team) consisting of 2 teachers and 3 students be formed if 6 teachers and 10 students are eligible to serve on the committee? |