CH 11.1 Permutations and Combinations TEACHER Notes.notebook
January 10, 2017
ALG 2 CH 11‐1 Permutations and Combinations Notes GOAL: Count permutations and combinations with at least 70% accuracy. What is a permutation? Give me a permutation of each set: (a) (b) Write ALL possible permutations of 5, 6, and 9: There are ____ permutations of the numbers 5, 6, and 9.
Sometimes listing and counting all the options is impractical; fortunately you can use the Fundamental Counting Principle:
IF ________ * _________ THEN there are M*N possibilities.
Event M can Event N can occur “N” ways. occur “M” ways.
Use the Fundamental Counting Principle when given x choices of A, y choices of B, and z choices of C: EX 1.1: How many ways to choose from 31 ice cream flavors, 4 cone types, and 7 toppings? 1
CH 11.1 Permutations and Combinations TEACHER Notes.notebook
January 10, 2017
EX 1.2: An old website requires 4‐character passwords consisting of 3 numbers and 1 letter. A new web site requires a 6‐character password consisting of 3 numbers and 3 letters. How many more passwords can be made for the new website? Old
New Look at Ch 11‐1 SOLVE IT (page 674):
1. You can list the options and look for a pattern:
Here are the combinations for Hamburger (H); we would need to create similar combinations for each sandwich type: HpA
HpB
HpF
HpR
HbA
HbB
HbF
HbR
HcA HcB HcF
HcR
There are 4 sandwiches TYPES with 12 options for each sandwich so there are 48 possible combinations. 2. OR you can use the Fundamental Counting Principle: __________ *__________ * __________ (# of sandwiches * # of sides * # of desserts)
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CH 11.1 Permutations and Combinations TEACHER Notes.notebook
January 10, 2017
RECALL: A ___________________________ is an arrangement of items in a particular order. Suppose that you wanted to find the number of ways to order three items. • There are ___ ways to choose the 1st item, ____ ways to choose the 2nd, and ____ way to choose the 3rd. • By the Fundamental Counting Principle, there are __________________________________ permutations. Using factorial notation, you can write 3 * 2 * 1 as 3!, read “three factorial.” • For any positive integer n, n factorial is n! à • Zero factorial is 0! = 1 ﴾you will show why this is true in your homework﴿
NOTE: A factoral is a product of all whole numbers less than or equal to a given number. Use factorals if asked to arrange x items in a row! EX 2 ‐ Finding the Number of Permutations of n items: In how many ways can you arrange nine CDs one after another on a shelf? ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
If you exchange the 1st and 2nd CD in the arrangement, would the result be a different permutation of the CDs? Explain. Sometimes you are interested in the number of permutations possible using all of the objects from a set, but just a few at a time. You can still use the Fundamental Counting Principle or factorial notation but you can also use a formula!
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Why can t r be greater than n? 3
CH 11.1 Permutations and Combinations TEACHER Notes.notebook
January 10, 2017
Suppose you want to rearrange the letters of the word “MATH” to form other four‐letter arrangements. How many are there? 1. You could like them and get 24 arrangements: 2. You could use the Fundamental Counting Principle: 3. OR … you could use the permutation formula where n=4 and r=4 (4 letters taken 4 at a time):
QUESTION: Is there an advantage to using either the Fundamental Counting Principle or the permutation formula when solving this problem? If asked to pick (choose) r items from n objects AND order matters use FCP (n)(n‐1)(n‐2) or EX 3 ‐ Finding In how many ways can a first, second, and third baseman be selected from eight players? 4
CH 11.1 Permutations and Combinations TEACHER Notes.notebook
January 10, 2017
If asked to pick (choose) r items from n objects AND order does NOT matter then use
EX 4.1 ‐ Finding Review QUESTIONS: • What is the difference between the formulas for permutations and combinations? • What is the value of ? Method 1
Method 2
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Is Explain. Evaluate 5
CH 11.1 Permutations and Combinations TEACHER Notes.notebook
January 10, 2017
When determining whether to use a permutation or combination, you must 1st decide whether order is important.
EX 5 ‐ Identifying Whether Order is Important
• List, or think about, a few possibilities to determine if order is important and to determine if there are duplicates (needed when order is NOT important, duplicates need to be divided out). • What would you use if order is important? •
• What would you use if order is NOT important?
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(5a) You have 20 songs on your music player. You have time to listen to four of the songs. How many ways can you listen to four of the songs? Does order matter? (5b) A raffle at a school carnival awards prizes of $100, $50, $25, and $10. If 25 raffle tickets are sold, how many different ways can the prizes be distributed? Does order matter?
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