Use Permutations to count Possibilities CCSS: 7.SP.8.b

1. 11.6 Use Permutations to count PossibilitiesCCSS: 7.SP.8.b

2. Vocabulary A permutation is an arrangement of objects in which is order is important.

3. Combinations and Permutations • What is the Difference? 1. We use the word “combination” loosely, without thinking if the order of things is important. In other words : “My fruit salad is a combination of peaches, bananas , apples”…..We don’t care about the order the fruits are placed because it is still a fruit salad.”

4. 2. “The combination to the safe was 689.” Now, we do care about the order. The safe would not open if we used “869” or “986”. It has to be exactly 6-8-9. if the order does not matter it is a Combination. If the order does matter it is a Permutation. Therefore all locks that require a code should be called Permutation locks. Video http://youtu.be/GmHGc8jZvIM

5. A permutation is an ordered combination. To help you remember, think Permutation……Position

6. For any positive integer n, the product of the integers from 1 to n is called n factorial and is written n!. The value of 0! is defined to be 1. What is a factorial? Watch the video to further explain. http://youtu.be/4j66DS_XTSo Watch the next video for further clarification if needed! http://www.virtualnerd.com/pre-algebra/probability-data-analysis/permutations-combinations/factorials/factorial-definition

7. Formula:Permutation = nPr = n! / (n-r)! Combination = nCr = nPr / r! where,              n, r are non negative integers and r<=n.              r is the size of each permutation.              n is the size of the set from which elements are permuted.              ! is the factorial operator.

8. Example 1: Counting Permutations There are 10 runners in a race. Assuming there are no ties, in how many orders can the runners finish? There are 10 possibilities for first place, 9 for second place, 8 for third place, 7 for fourth place, 6 for fifth place, 5 for sixth place, 4 for seventh place, 3 for eighth place, 2 for ninth place, and 1 for tenth place. So, the number of orders in which the runners can finish is 10! 10! = 10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 3, 628, 800 There are 3, 628, 800 orders in which the runners can finish.

9. Evaluate the factorial Let’s Practice 1. 4! Hint: (4 · 3 · 2 · 1) Answer: 24 8! Answer: 40, 320 3. 9! Answer: 362, 880 4. 0! = Yes…….it equals 1

10. Permutations • Algebra The number of permutations of n objects taken r at a time can be written as • n P r Where n Pr Numbers: 5 P 3 = = = Once you simplify [5 x 4 x 3 ] = 60

11. Example 2: Counting Permutations • You have 8 CD’s, but you only have a 5 – disc CD changer. How many different arrangements of the CD’s are possible to put in the CD changer. • Solution: To find the number of ways that the CD’s can be put in the CD changer, find 8 P5. • 8P5 = = = = • 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 6, 720 • There are 6,720 ways the CD’s can be put in the CD changer.

12. Find the number of permutations • 1. 10P5 Answer: 30, 240 • 7P3 Answer: 210

13. Example 3: Finding a probability using permutations • You know the first 3 digits of your new friend’s phone number. You also know that the last four digits are 0, 5, 8, 9, but you can’t remember the order of the last four digits. You randomly pick an order for the last 4 digits. What is the probability that you correctly guess the phone number.

14. Solution Each possible arrangement of the last four digits of the phone number is a permutations of the digits 0, 5, 8, 9. The number of permuations of the last four digits is 4! 4! = 4· 3 · 2 · 1 = 24 Only one of the possible permutations is correct, so the probability of correctly guessing the phone number in one attempt is .

15. Video: [Just for fun]http://youtu.be/BbX44YSsQ2I 1. http://youtu.be/-mC_QK6dBIY • http://learni.st/users/S33572/boards/3593-permutations-combinations-and-probability-common-core-standard-9-12-s-cp-9 Compass Learning: Folder labeled Permutations] you may stop the video when combinations begins. Print score and attach your work: Due 2/13