X "Waiting for response fr X S Live Discussion - Studypool + edu.americanhighschool.org/dashboards/ActiveLearning/course-website.asp?action=assignment&Submitid=27113&srloid=27113&urloid=27113&courseid=1486 O☆ = 2 American High School Dashboard * Management v O Reports Cassidy Williams Assessments Unit 9: Applications of Probability Mid Checkpoint Apply Unit 9: Mid Checkpoint Apply Interactive Tools Unit 9: Applications of Probability Mid Checkpoint Apply Lesson 1 - 3 Please complete the following questions. It is important that you show all work you did to solve the problems when you submit your work. This includes any calculations, diagrams, or graphs that helped you solve it. Algebra 2 1. OPEN ENDED Describe a situation in which the number of outcomes is given by 26,3). 2. OPEN ENDED Describe an event that has a probability of O and an event that has a probability of 1. 3. Writing in Math Use the information on page 684 to explain how you can count the maximum number of license plates a state can issue. Explain how to use the Fundamental Counting Principle to find the number of different license plates in a state such as Oklahoma, which has 3 letters followed by 3 numbers. Also explain how a state can increase the number of possible plates Without increasing the length of the plate number. 4. Writing in Math Use the information on page 710 to explain how permutations and combinations apply to softball. Explain how to find the number of 9-person lineups that are possible and how many ways there are to choose 9 players if 16 players show up for a game. Click here to chat Type here to search g 3:32 PM 8/17/2017 19 x "Waiting for response from S Ask a new question - Study CC - Algebra 2 + ə intervisualtechnology.us/uploads/PDFs/ebooks/CC%20-%20Algebra%202/CC%20-%20Algebra%202 U ☆ = Write two more rows of Pascal's triangle. Then use the patters of the cients Write American High School LA Main Ideas . Uwcastlange the and post binomas . Uw the inmal Theorm to espand powers of biomis New Vocabulary Paulstring Binomial Theorem factorial 684 - 685 / 1100 GET READY for the Lesson According to the US Census Bureau, ten percent of families have three or more children. Il a family has four children, there are six sequences of births of boys and girls that result in two boys and two girls. These sequences are listed below DIGG TXGB GANG ССП GGDE (* + y = 1x?y + 7x*y+ 21x2 + xy 100% = x? +7xy + 2x +354 CHECK Your Progress 1. Expand () TCRC . The Binomial Theorem Another way slow cxpansion is to write them in terms of the previa (9+) -Pascal's Triangle You can use the coefficients in powers of binomials to count the number of possible sequences in situations such as the one above. Expand a few powers of the binomials + () + 8)0 - 147 15+ x) = 1+14 (1 + x)2 = 153 +211 + 1/2 1 +81' 1578" + 3871 +3818+10 () + 19 - 1543" this' ++48 g + 1 1 2:1 (a+b 1 ² 루 2 1:3 3.2.1 (+) 1 4 4.3.2 Study Tip This pattern summarized in the Binomial Theorem Terms The open of : Drama teha po 2-1 Sorms. For enke 10 by has 7 term. KEY CONCEPT Binomial Theorem in is a nonnegative integer, then (a + b)" - 10"-16 2 g 32+ " 207 393 + ... + 10% The ethicient 4 of the big term in the expansion of (b + 1 gives the number of sequences of births that result in one boy and three girls Here are some patterns in any binomial expansion of the form (a + b)". 1. There are n + 1 terms. 2. The expectent of a + " is the exponent of a in the first term and the exponent of it in the last team 3. In successive terms, the exponent of a decreases by one, and the exponent of increases by one. 4. The sum of the exponents in each terris 5. The coefficients are symmetric. They increase at the beginning of the expansion and decrease at the end The coefficients form a pattern that is often displayed in a triangular tormation. This is known as Pascal's triangle. Notice that each row begins and ends with 1. Each coefficient is the sum of the two coefficients above it in the previous row. a + b) 1 + b) 1a + b)2 Real World Unh Nthough he ad not GREAT I, Pascats trangle is named for the Free mathematician Blaise Pascal (1623-15621. Study Tip Coefficients Matat ns haaing the same esponse reverse, as in 15 and 1500 EXAMPLE Use the Binomial Theorem Expand la-6) Use the sequence 1, 11:9.5 to find the coefficients for the first four terms. Then use symmetry to find the remaining cetticients. 19 – by® = 1a* (_biº + *** 1-by! - 6:3* (_h=:5**(-1) + + 1-2 gh-61 +15.12-2017 152746 CHECK Your Progress 2. mandir . F Type here to search g 9 سا 3:34 PM 8/17/2017 719 x "Waiting for response from S Ask a new question - Study CC - Algebra 2 + o с intervisualtechnology.us/uploads/PDFs/ebooks/CC%20-%20Algebra%202/CC%20-%20Algebra%202 ☆ = yQ Simply 710 - 711 / 1100 Mobie that in amg e factors are al of Louan a cause the Expression in the kolina American High School 20 by common laci Main Ideas . Suke problems Prmindful Sake prins iwing cominans (ID - 3 - GET READY for the Lesson When the manager of a softball team ills out her team's lineup card before the game, the order in which she fills in the names is important because it determines the order in which the players will bat. Suppone she has 7 posible players in mind for the top 4 spots in the lineup. You know from the fundamental Counting Principle that there are 7.6.5.4 or 540 ways that she could assign players to the top 4 spots The gold, silver, and bronze medas can b4100%nded in 20 wa CHECK Your Progress 1. A newspaper has nine reporters available How many ways can the texts be assig -9.9.20 New Vocabulary permutation Suppose you want to rearrange the tters of the can make a different word. If the two es were the word could le arranged in P 8.8 ways. The divide 48,8) by the number of arrangements in P(2.2) ways combination 18,81 S! M12, 273 8.7.6-5.-1.3.2! or 20,160 Simplify Thus, there are 27,160 ways to arrange the Ictters in gemetry. Permutations When a group of objects or people are arranged in a certain order the arrangement is called a permutation. In a permutation, the order of the objects is very important. The arrangement of objects or prople in a line is called a lincar permutation. Notice that 7.6.5.4 is the product of the first 4 factors of 71. You can rewrite this product in terms of 1 7.6.5.4 = 7.6.3.4.3.2.1 3.2.1 When some letters or objects are alike, we the rule below to find the num of permutations KEY CONCEPT Permutations with Repetiti The number of permutations of n objects of which p are alike and gare alike is = 7.6-5.4:3.2.1 or 21-7-6-5-4-3-2-1 and 31 -3.2.1 pligt This rule can be extended to any number of objeds that are repeated Notice that is the same 17-4) The number of ways to arrange 7 people or objects taken 4 at a time is written "[7, 4). The expression for the sotthall lineup above is a case of the following formula KEY CONCEPT Permutations The number of pemutations of a distinct objects taken at a time is given by Pinn) - ni (n-1) Reading Math Permutations the spression in reads the number of PAVOL culars tolawa be is sometimes EXAMPLE Permutation with Repetition How many different ways can the letters of the word MISSISSIPPI be arranged? The letter occurs 4 times, Soccurs 4 times, and P occurs wie You need to find the number of permutations of 11 letters of which 4 of one letter, 4 of another latter, and 2 of another Ictter and the same. 11.10.9.4.2.6.5.40 34,650 1121 There are 34,650 ways arrange the letters. CHECK Your Progress 2. How many different ways can the letters of the word DECIDED be weena EXAMPLE Permutation FIGURE SKATING There are 10 finalists in a figure skating competition. How many ways can gold, silver and bronze medals be awarded? Since each winner will recive a diferent medal ander is important. You E Type here to search (X 3:34 PM 8/17/2017 719
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