Permutations
 4! is read as 4 factorialand is...4 x 3 x 2 x 1.  It equals24.3! = 5! is read as 5 factorialand is...5 x 4 x 3 x 2 x 1.  It equals 120.7! = 7 x 6 x 5 x     ••• 3! =         x        x  0! is read as 0 factorialand equals 1. 3! is read as 3 factorialand is...3 x 2 x 1.  It equals 6.4! =       x      x     x Multiplication Principle: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:  =n × m × p × ... Example1You 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.xx= Example2You have three pairs of jeans, five t-shirts, and two pairs of shoes.  Determine the number of possible ways to dress up.xx= I have two different rings, fourhats, and three jackets. How many ways can I select a ring,hat, and jacket to wear?9 241/24 12  Susan has 4 colors of wrapping paper, 3 colors of ribbon, and 2 types of tape. How many different ways does she have to wrap a package with one color of wrapping paper, one color of ribbon, and one type of tape?141224 Create a tree diagram for the following problem. An ice cream shop has vanilla andchocolate ice cream, and chocolate and strawberry syrup.  You can also choosebetween sprinkles or nuts. Make a tree diagram showing the outcomes.Ice Cream Flavor     Syrup Flavor                Toppingvanillachocolatechocolatestrawberrynutssprinklessprinklesnuts PermutationsLet E be a set containing n elements ( Card(E) =n ), Let  rbe a natural number  ( r ≤ n). Then a permutation without repetition of the r elements selected from E is any ordered sequence taken from the elements of E containing exactly r distinct elements(No repeated element in a single permutation)Eg: Let E= {1,2 ,3 ,4,5,6} then (1,3,5) is a permutation of E containing 3 elements, (1,5,3) is another permutation of E containing 3 elements, (1,2) is a permutation of E containing two element and (652413) is a permutation of E containing 6 elements>Note1: (1,1,3) and (2,2) are not permutations without repetitions> Note2: Permutations without repetitions are simply called permutations Permutations - order matters no repetition  Of n things taken r at a time.nPr=           orn count down r times(n-r)!n! N is the number of items being arranged in groups of r elements each.  How many items are there to pick from ?nPr=           orn count down r times(n-r)!n! N is the number of items being arranged in lots of r.  How many items are in each group?         itemsnPr=           orn count down r times(n-r)!n! For this problem start at 6 and count down two place.Permutations - order matters no repetition   Of n things taken r at a time.nPr= 6 x 5n multiplied by count down r places(n-r)!n!or6P2= (6-2)!6! Permutations - order matters no repetition  Of n things taken r at a time.nPr6 x 5 x 4 ==                  orn count down r times(n-r)!n!6P3= (6-3)!6! Permutations - order matters no repetition  Of n things taken r at a time.nPr7 x 6 x 5 ==                 orn count down r times(n-r)!n!7P3= (7-3)!7! Permutations - order matters no repetition  Of n things taken r at a time.8P3   x     x      =  336= count down 3 spots(8-3)!8!= 8!! Ex:1 How many anagrams are there of the letter SEA?(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 The possible cases are :  SEA, SAE, ESA, EAS, ASE , AES Ex:2Four horses are running a race.  How manydifferent ways can these horses come in first, second, and third? You forget the pin number for your computer which consists of 4 distinct digits. You remember that the firstdigit is 1 and the remaining three digits are all even number.What is the maximum number of attempts before you can find the correct pin and unlock your computer?RuleAnswer r-listsRepetition 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 placeRepetition not allowed - order importantRepetition Allowed - order important r-lists?Permutation? Permutationr-listsnr?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 aren=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?Solutionwe have to choose 3 elements from the letters {H, T}The order is important and we must have repetitions. So: Then nr=n=andr= 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=8HTTHHT To find all the previous outcomes, we use a Tree-diagram: Ex:1In 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:2×= Ex: 3How many four-digit numbers can be formed? Ex: 4How many four digit numbers can be formed with distinct digits? Ex: 5How many four digit numbers can be formed less than 5000? Ex: 6How 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: 7×= A symbol can be an English letter (Upper or lower case), a digit from 0 to 9 or one of : %, \$, @, -, _, # You Know that the password for your account is formed of 6 symbols.How many possible passwords are there? ( Note: the letters are case: sensitive: means that A≠a )Ex:8Rule RuleEx:9Answer RuleAnswer Note:If you want something AND something use     (×) Note:If you want something OR something use     (+)
Students who took this test also took :

Created with That Quiz — where test making and test taking are made easy for math and other subject areas.