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2 Warm Up Tell whether each event is dependent or independent 1. Selecting a name from a Chicago telephone book and a name from the Houston telephone book 2. Tossing a coin twice 3.Choosing two cards from a deck so that they make a “pair” (the number value is the same)

Copyright © Ed2Net Learning Inc.3 Warm Up Find the probability 4. Two dice are rolled. Find the probability that a multiple of three is rolled on one die and an even number is rolled on the second die. 5. A blue number cube and a yellow number cube are rolled. Find the probability that an odd number is rolled on the blue number cube and a multiple of 6 is rolled on the yellow number cube.

Copyright © Ed2Net Learning Inc.4 Lets review what we learned in the last lesson If A and B are dependent events, the probability of both events occurring is the product of the probability of the first event and the probability of the second event once the first event has occurred. P(A and B) = P(A) x P(B, once A has occurred ) Dependent events Two events, A and B, are dependent if the fact that A occurs does not affects the probability of B occurring. Examples: Choosing a ball from a box AND choosing another ball from that box.

Copyright © Ed2Net Learning Inc.5 Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events P(A and B) = P(A) x P(B). Independent events Choosing a ball from a box AND landing on tails after tossing a coin. Review of the last lesson Examples:

Copyright © Ed2Net Learning Inc.6 Fundamental Counting Principle Simple counting problems allow one to list each possible way that an event can occur. However, some events can occur in so many different ways that it would be difficult to write out an entire list. Hence, one must use the Basic counting principle:The mathematical theory of counting is known as combinatorial analysis. Suppose a task involves a sequence of r choices. Let n1 be the number of ways the first stage or event can occur and n2 be the number of ways the second stage or event can occur after the first stage has occurred. Lets get Started

Copyright © Ed2Net Learning Inc.7 Continuing in this way, let nr be the number of ways the rth stage or event can occur after the first r - 1 stages or events have occurred. Then the total number of different ways the task can occur is: n1 * n2 * n3 * ………………… * nr A factorial notation is used for writing the product of all the positive whole numbers up to a given number.

Copyright © Ed2Net Learning Inc.8 Factorials In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. This is written as n! and pronounced 'n factorial‘. The numeric value of n! can be calculated by repeated multiplication, if n is not too large. The Fundamental Counting Principle says that the number of ways to choose n items is n ! n! = n * (n-1) * (n-2) *……………*3 * 2 * 1

Copyright © Ed2Net Learning Inc.9 For example, 4! = 4 * 3* 2* 1 = 24 0! = 1 because the product of no numbers at all is 1 (This is a convention) Calculations with factorials are based on the fact that, any factorial less than n! is a factor of n! For example: 6! is a factor of 10!. 10! = 1· 2· 3· 4· 5· 6· 7· 8· 9· 10 = 6!· 7· 8· 9·

Copyright © Ed2Net Learning Inc.10 Factorials - examples If we multiply both the terms on the left by n, then Example 1: Evaluate 9! 5! 9! 5! = = = ! 5! Example 2: Show 1! (n-1)! = n n! 1! (n-1)! = n (n-1)! n Now, (n − 1)! is the product up to the number just before n. Therefore, (n − 1)! n itself is n!. 1! (n-1)! = n n!

Copyright © Ed2Net Learning Inc.11 Factorials - Application Factorials are important in combinatorial analysis. For example, there are n! different ways of arranging n distinct objects in a sequence. These arrangements are called permutations and the number of ways one can choose r objects from among a given set of n objects. These arrangements are called combinations.

Copyright © Ed2Net Learning Inc.12 Combinations A combination is defined as a possible selection of a certain number of objects taken from a group without regard to order. For example, suppose you want to choose 2 letters from a group of 4 letters. If the group of letters are P, Q, R and S, we can choose the letters in combinations of two as follows: PQ, PR, PS, QR, QS, RS The order in which the letters appear doesn’t matter here. I.e PQ can be written as QP, PR as RP, etc. So, there are 6 combinations while choosing two letters from a group of four letters.

Copyright © Ed2Net Learning Inc.13 Permutation and Combination - Difference The distinction between a combination and a permutation has to do with the sequence or order in which objects appear. A combination focuses on the selection of objects without regard to the order in which they are selected. A permutation, in contrast, focuses on the arrangement of objects with regard to the order in which they are arranged.

Copyright © Ed2Net Learning Inc.14 Representing a combination A combination of n objects taken r at a time is a selection which does not take into account the arrangement of the objects. That is, the order is not important. The number of ways (or combinations) in which r objects can be selected from a set of n objects, where repetition is not allowed, is denoted by, or C(n, r) 3 5 C n C r = n! (n-r)! r! Example 1:Find the value of = C n C r = 5! (5-3)! 3! 3 5 C = 2! * 3! 5 * 4 * 3 * 2 * 1 5! = (2 * 1 ) ( 3 * 2 * 1)

Copyright © Ed2Net Learning Inc.15 Properties of Combinations 1) C(n, r) = C(n, n-r). [symmetric nature of a combination] Example: C(10, 7)= 10!/3! 7! is the same as C(10, 3) C(10, 3) = 10!/ 7! 3! 2) C(n, 1) = n Example: C(3, 1) = 3!/ 2! 1! = 3 3) C(n, 0) = 1 Example: C(5, 0) = 5!/(5-0)! 0! = 5!/5! = 1 4) C(n, n)=1 Example: C(4, 4) = 4!/ (4-4)! 4! = 4!/4! = 1 5) The sum of all combinations : C(n,0) + C(n, 1) C(n, n) = 2^n

Copyright © Ed2Net Learning Inc.16 Pascal’s Triangle Pascal’s Triangle: Pascal’s Triangle illustrates the symmetric nature of a combination. i,e C(n,r) = C ( n, n - r) Combinations are used in the binomial expansion theorem from algebra to give the coefficients of the expansion (a+b)^n.They also form a pattern known as Pascal’s Triangle.

Copyright © Ed2Net Learning Inc Each element in the table is the sum of the two elements directly above it. Each element is also a combination. C (n, r) = C (n, n - r) The n value is the number of the row (start counting at zero) and the r value is the element in the row (start counting at zero)

Copyright © Ed2Net Learning Inc.18 Relation between permutation and combination How are the number of combinations related to the number of permutations, ? To explain this, let us consider an example. Consider the selection of 3 alphabets from a group of 4 alphabets say A, B, C and D. All the combinations of A, B, C and D taken 3 at a time are, ABC, ABD, ACD, BCD These four combinations of 4 things taken 3 at a time, is denoted as. Since the order does not matter in combinations, there are clearly fewer combinations than permutations. The combinations are contained among the permutations.They are a subset of the permutations. Each of those four combinations, will give rise to 3! i.e, 6 Permutations. …continued 4 C 3 p n r n C r

Copyright © Ed2Net Learning Inc.19 Relation between permutation and combination The permutations are, ABC ABDACDBCD ACBADBADCBDC BACBADCADCBD BCABDACDACDB CABDABDACDBC CBADBADCADCB Each column is the 3! permutations of that combination. But they are all one combination, as the order does not matter. Hence there are 3! times as many permutations as combinations i.e in total 24 permutations. Combination, therefore, will be permutation divided by 3!. So, 4 C 3 p 4 3 p C 3 = 3! In general, p n r n C r = r! n C r = n(n-1)(n-2) ………r factors r!

Copyright © Ed2Net Learning Inc.20 Examples 1. 1.In how many ways could you select three of these digits: 1, 2, 3, 4, 5 ? There are 5 digits. Three digits have to be selected. So the combination is 5 C 3 5 C 3 = 5! (5-3)! 3! = 10 5 C 3 2. The Florida Lottery is the biggest single state lottery in the country. In the game you must choose six numbers between one and fifty three. If you match them all, you win! In how many ways can you pick six numbers? Here n = 53, r = 6, 53 C 6 = ? 53 C 6 = 53! (53-6)! 6! = 22,957,480 So, there are 22,957,480 ways to choose 6 numbers.

Copyright © Ed2Net Learning Inc.21 Examples In how many ways can 5 letters be posted in 3 post boxes, if any number of letters can be posted in all of the three post boxes? The first letter can be posted in any of the 3 post boxes. Therefore, it has 3 choices. Similarly, the second, the third, the fourth and the fifth letter can each be posted in any of the 3 post boxes. Therefore, the total number of ways the 5 letters can be posted in 3 boxes is, 3 to the power 5 = 3*3*3*3*3 (3 to the power 5) = 243 ways.

Copyright © Ed2Net Learning Inc.22 Your Turn 1. 1.How many 4-element subsets can be formed from the set, {a,b,c,d,e,f,g}? 2. Ms. Nancy will choose 1 boy and 1 girl from her class to be the class representatives. If there are 3 boys and 7 girls in her class, how many different pairs of class representatives could she pick? (Hint: The number of combinations is the product of the number of things you have to choose from) 3. 3.Kim is going to buy an ice cream sundae. A sundae consists of 1 flavor of ice cream and 1 topping. If she can choose from the kinds of ice cream and toppings below, how many different sundaes could she create?

Copyright © Ed2Net Learning Inc.23 Your Turn 4. 4.How many ways can you arrange 6 different books, so that a specific book is on the third place? 5. In how many ways can 7 children be seated at a circular table? 6. In a conference of 9 schools, how many intra conference football games are played during the season if the teams all play each other exactly once? 7. 8 students names will be drawn at random from a hat containing 14 freshmen names, 15 sophomore names, 8 junior names, and 10 senior names. a) How many different draws of 8 names are there overall? b) How many different draws of 8 names would contain only juniors?

Copyright © Ed2Net Learning Inc.24 Your Turn 8. 8.How many committees consisting of 3 teachers and 5 students can be made up if there are 6 teachers and 10 students to choose from? 9. 9.The Atlanta Braves are having a walk-on tryout camp for baseball players. Thirty players show up at camp, but the coaches can choose only four. How many ways can four players be chosen from the 30 that have shown up? 10. Five-card stud is a poker game, in which a player is dealt 5 cards from an ordinary deck of 52 playing cards. How many distinct poker hands could be dealt?

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Copyright © Ed2Net Learning Inc How many different 7-digit telephone numbers are possible if the first digit can not be zero and no digit may repeat?

Copyright © Ed2Net Learning Inc Seven people take part in a panel discussion. Each person is to shake hands with all of the other participants at the beginning of the discussion. How many handshakes take place? List them all.

Copyright © Ed2Net Learning Inc.29 3.A door can be opened only with a security code that consists of five buttons: 1, 2, 3, 4, 5. A code consists of pressing any one button, or any two, or any three, or any four, or all five. a) How many possible codes are there? b) If, to open the door you must press three codes, then how many possible ways are there to open the door? Assume that the same code may be repeated

Copyright © Ed2Net Learning Inc.30 Let’s summarize what we have learnt today 1. The fundamental counting principle says that, the the total number of different ways a task can occur is: n1 * n2 * n3 * ………………… * nr where n1, n2, n3..nr are the tasks performed at first, second and rth stages 2. Factorial is a shorthand notation for a multiplication process. The Fundamental Counting Principle says that the number of ways to choose n items is n ! n! = n * (n-1) * (n-2) *……………*3 * 2 * 1 3. Any factorial less than n! is a factor of n! 4. A combination focuses on the selection of objects without regard to the order in which they are selected.

Copyright © Ed2Net Learning Inc A permutation in contrast, focuses on the arrangement of objects with regard to the order in which they are arranged. 6. The number of ways (or combinations) in which r objects can be selected from a set of n objects, where repetition is not allowed, is denoted by, or C(n, r) n C r n! n C r = (n-r)! r! 7. Pascal’s Triangle illustrates the symmetric nature of a combination. i,e C(n,r) = C(n,n-r) Recap

Copyright © Ed2Net Learning Inc The sum of all combinations : C(n,0) + C(n, 1) C(n, n) = 2^n 12. Relationship between permutation and combination is p n r n C r = r! Recap 8. 8.C(n, 1) = n 9. 9.C(n, 0) = C(n, n) = 1

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