How much should I invest? I got the perfect investment offer. For the last 10 weeks I received a share recommendation from a fund manager, telling me whether a stock’s price would rise or fall over the next week. After ten weeks, all the recommendations were proved right. So, he predicted the future 10 times in a row. There is only a one-in-a-thousand chance that the result is down to luck. I think this guy is a genius. I plan to invest all my kids’ college money. What do you think?

The Math  

This is a well-known scam This is a well-known scam. The promoter sends out 100,000 e-mails, picking a stock at random. Half the recipients are told that the stock will rise; half that it will fall. After the first week, the 50,000 who received the successful recommendation will get a second e-mail; those that received the wrong information will be dropped from the list. And so on for ten weeks. At the end of the period, just by the law of averages, there should be 98 punters convinced of the manager’s genius and ready to entrust their savings

The Math  

WARMUP Lesson 10.1, For use with pages 682-689 Evaluate the expression. Hint: NO Calculator 1. 5 4 3 2 1 ANSWER 120 2. 7 6 5 4 3 2 1 6 5 4 3 2 1 2 3 ANSWER 7/6 3. The number of choices Meghan has for displaying her trophies is represented by the expression a • b. If a = 4 and b = 3, how many choices does Meghan have? ANSWER 12 4. How many different 4-digit bank pin numbers are there? ANSWER 10,000

10.1 Notes - Counting Principal & Permutations Fictitious Vampires Duke it Out

Objective -To count the number of ways an event can happen.

How many possible 7-digit phone numbers are there if the first digit cannot be 0 or 1?

A multiple choice test has 6 questions with 5 choices each. In how many ways can you complete the test?

3 4 3 5 5 2 How many three-digit numbers greater than 500 can be formed from the digits 1, 2, 5, 7 and 9 if no repetition is allowed? 3 4 3 5,7,9 How many positive three-digit even numbers can be formed without using the digits 0, 1, 2, 3 or 4? 5 5 2 5-9 5-9 6,8

A certain state has license plates that are made of four numbers followed by two letters? How many different license plates are possible if no repeated letters or numbers are allowed?

!

Say “Factorial” When you see “. ”, say Factorial Say “Factorial” When you see “!”, say Factorial. It means to multiply that number by all the positive integers less than it. Your calculator has a “button” for it. I suggest you find it. Graphing calculators, look under “Math”

Permutation- is an ordering of n objects. How many permutations are there for the letters in the word CAT? CAT, CTA, ACT, ATC, TCA, TAC = 6 How many permutations are there for the letters in the word CAT?

There are 9 songs on your MP3 player There are 9 songs on your MP3 player. In how many different ways could you listen to these songs? If you only had time to listen to 3 of the 9 songs on your MP3 player, how many different ways could you listen to these 3 songs?

In how many different ways can the letters in the word TRIANGLE be arranged? How many different ways could 10 runners finish in first, second and third place?

Find the number of permutations of 7 objects taken 4 at a time.

Find the number of permutations for the letters in the word STREET.

Find the number of permutations for the letters in the word MATHEMATICS. Find the number of permutations for the letters in the word SLEEPLESS.

By Definition… Write it down!!!! 0! = 1

Do this one!!!! EXAMPLE 1 Use a tree diagram Snowboarding A sporting goods store offers 3 types of snowboards (all-mountain, freestyle, and carving) and 2 types of boots (soft and hybrid). How many choices does the store offer for snowboarding equipment? SOLUTION Draw a tree diagram and count the number of branches.

EXAMPLE 1 Use a tree diagram ANSWER The tree has 6 branches. So, there are 6 possible choices.

EXAMPLE 2 Use the fundamental counting principle Photography You are framing a picture. The frames are available in 12 different styles. Each style is available in 55 different colors. You also want blue mat board, which is available in 11 different shades of blue. How many different ways can you frame the picture?

EXAMPLE 2 Use the fundamental counting principle SOLUTION You can use the fundamental counting principle to find the total number of ways to frame the picture. Multiply the number of frame styles (12), the number of frame colors (55), and the number of mat boards (11). Number of ways = 12 55 11 = 7260 ANSWER The number of different ways you can frame the picture is 7260.

EXAMPLE 3 Use the counting principle with repetition License Plates The standard configuration for a Texas license plate is 1 letter followed by 2 digits followed by 3 letters. How many different license plates are possible if letters and digits can be repeated? How many different license plates are possible if letters and digits cannot be repeated?

EXAMPLE 3 Use the counting principle with repetition SOLUTION There are 26 choices for each letter and 10 choices for each digit. You can use the fundamental counting principle to find the number of different plates. Number of plates = 26 10 10 26 26 26 = 45,697,600 ANSWER With repetition, the number of different license plates is 45,697,600.

EXAMPLE 3 Use the counting principle with repetition If you cannot repeat letters there are still 26 choices for the first letter, but then only 25 remaining choices for the second letter, 24 choices for the third letter, and 23 choices for the fourth letter. Similarly, there are 10 choices for the first digit and 9 choices for the second digit. You can use the fundamental counting principle to find the number of different plates. Number of plates = 26 10 9 25 24 23 = 32,292,000 ANSWER Without repetition, the number of different license plates is 32,292,000.

GUIDED PRACTICE for Examples 1, 2 and 3 SPORTING GOODS The store in Example 1 also offers 3 different types of bicycles (mountain, racing, and BMX) and 3 different wheel sizes (20 in., 22 in., and 24 in.). How many bicycle choices does the store offer? 9 bicycles ANSWER

GUIDED PRACTICE for Examples 1, 2 and 3 WHAT IF? In Example 3, how do the answers change for the standard configuration of a New York license plate, which is 3 letters followed by 4 numbers? ANSWER The number of plates would increase to 175,760,000. The number of plates would increase to 78,624,000.

EXAMPLE 4 Find the number of permutations Olympics Ten teams are competing in the final round of the Olympic four-person bobsledding competition. In how many different ways can the bobsledding teams finish the competition? (Assume there are no ties.) In how many different ways can 3 of the bobsledding teams finish first, second, and third to win the gold, silver, and bronze medals?

EXAMPLE 4 Find the number of permutations SOLUTION There are 10! different ways that the teams can finish the competition. 10! = 10 9 8 7 6 5 4 3 2 1 = 3,628,800 Any of the 10 teams can finish first, then any of the remaining 9 teams can finish second, and finally any of the remaining 8 teams can finish third. So, the number of ways that the teams can win the medals is: 10 9 8 = 720

GUIDED PRACTICE for Example 4 WHAT IF? In Example 4, how would the answers change if there were 12 bobsledding teams competing in the final round of the competition? The number of ways to finish would increase to 479,001,600. ANSWER The number of ways to finish would increase to 1320.

EXAMPLE 5 Find permutations of n objects taken r at a time Music You are burning a demo CD for your band. Your band has 12 songs stored on your computer. However, you want to put only 4 songs on the demo CD. In how many orders can you burn 4 of the 12 songs onto the CD? SOLUTION Find the number of permutations of 12 objects taken 4 at a time. ( 12 – 4 )! 12P4 = 12! = 11,880 8! 479,001,600 40,320 ANSWER You can burn 4 of the 12 songs in 11,880 different orders.

GUIDED PRACTICE for Example 5 Find the number of permutations. 5P3 = 60 ANSWER 4P1 = 4 ANSWER 8P5 = 6720 ANSWER

GUIDED PRACTICE for Example 5 Find the number of permutations. 12P7 = 3,991,680 ANSWER

EXAMPLE 6 Find permutations with repetition Find the number of distinguishable permutations of the letters in MIAMI and TALLAHASSEE. SOLUTION MIAMI has 5 letters of which M and I are each repeated 2 times. So, the number of distinguishable permutations is: = 30 2! 2! 5! = 120 2 2

EXAMPLE 6 Find permutations with repetition TALLAHASSEE has 11 letters of which A is repeated 3 times, and L, S, and E are each repeated 2 times. So, the number of distinguishable permutations is: 3! 2! 2! 2! 11! = 39,916,800 6 2 2 2 = 831,600

GUIDED PRACTICE for Example 6 Find the number of distinguishable permutations of the letters in the word. MALL 12 ANSWER

GUIDED PRACTICE for Example 6 Find the number of distinguishable permutations of the letters in the word. KAYAK 30 ANSWER

GUIDED PRACTICE for Example 6 Find the number of distinguishable permutations of the letters in the word. CINCINNATI 50,400 ANSWER

10.1 Assignment 10.1: 3 OR 5 (Tree Diagrams – look at example 1 in the book and check your answer in the back), 13-53 EOO, 65 (WP in the back), 73,75 Team Count!!