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4! is read as 4 factorial and is... 4 x 3 x 2 x 1. It equals 24. 3! = 5! is read as 5 factorial and 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 factorial and equals 1. 3! is read as 3 factorial and 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 × ... 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 = I have two different rings, four hats, and three jackets. How many ways can I select a ring, hat, and jacket to wear? 9 24 1/24
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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? 1 4 12 24 Create a tree diagram for the following problem. An ice cream shop has vanilla and chocolate ice cream, and chocolate and strawberry syrup. You can also choose between sprinkles or nuts. Make a tree diagram showing the outcomes. Ice Cream Flavor Syrup Flavor Topping vanilla chocolate chocolate strawberry nuts sprinkles sprinkles nuts Permutations Let 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. n P r = or n 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 ? n P r =
or n 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? items n P r =
or n 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. n P r = 6 x 5 n multiplied by count down r places (n-r)! n! or 6 P 2 = (6-2)! 6! Permutations - order matters no repetition Of n things taken r at a time. n P r 6 x 5 x 4 = = or n count down r times (n-r)! n! 6 P 3 = (6-3)! 6! Permutations - order matters no repetition Of n things taken r at a time. n P r 7 x 6 x 5 = = or n count down r times (n-r)! n! 7 P 3 = (7-3)! 7! Permutations - order matters no repetition Of n things taken r at a time. 8 P 3 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:2 Four 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? Rule Answer 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 Allowed - order important r-lists ? Permutation ? Permutation r-lists nr ? 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: Ex:1 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:2 × = Ex: 3 How many four-digit numbers can be formed? Ex: 4 How many four digit numbers can be formed with distinct digits? Ex: 5 How many four digit numbers can be formed less than 5000? Ex: 6 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: 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:8 Rule Rule Ex:9 Answer Rule Answer Note:If you want something AND something use (×) Note:If you want something OR something use (+) |