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n Factorial & Permutations

n Factorial & Permutations. What is n Factorial?. “The factorial of n is denoted by n! and calculated by the product of integer numbers from 1 to n” For n>0, n ! = 1×2×3×4×...× n For n=0 , 0 ! = 1. Huh? . Here are some examples…. 1! = 1 2! = 2x1= 2 3! = 3x2x1 = 6

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n Factorial & Permutations

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  1. n Factorial & Permutations

  2. What is n Factorial? “The factorial of n is denoted by n! and calculated by the product of integer numbers from 1 to n” • For n>0, n! = 1×2×3×4×...×n • For n=0, 0! = 1 Huh?

  3. Here are some examples… • 1! = 1 • 2! = 2x1= 2 • 3! = 3x2x1 = 6 • 4! = 4x3x2x1 = 24 • 5! = 5x4x3x2x1 = 120 You simply multiply the numbers of whichever “n” you have. 

  4. The Door Lock Problem • Sherlock Holmes is investigating a crime at a local office building after hours. In order to enter a building, he must guess the door code. • If only one number opens the door, how many different ways can Sherlock open the door?

  5. (write this down on your notes) 1 2 3 4 5 5 ways!

  6. (take 2-3 min to complete w/ a partner) • If two numbers will open the door, list the different combinations of buttons that would open the door. (Assume no repetitions, i.e 3,3 ) Example: (1, 2), (1, 3), (1, 4)… 1,2 1,3 1,4 1,5 2,1 2,3 2,4 2,5 3,1 3,2 3,4 3,5 4,1 4,2 4,3 4,5 5,1 5,2 5,3 5,4

  7. So… • How many ways could Sherlock open the door if two buttons will unlock it? 20 ways! 54 How many possible choices? 5 Now, how many possible choices? 4

  8. What if… • If three numbers will open the door, list the different combinations of buttons that would open the door. (Assume no repetitions) Example: (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 3, 5)… 1,2,4 1,2,5 … 1,2,3 1,3,2 There has to be another way…

  9. So then? • How many different ways could Sherlock open the door if three buttons will unlock it? 543 A grand total of 60 ways to unlock that door

  10. YES! • This does mean that if you now need to use FOUR buttons, it will be: • Again, the 5 does not mean you selected “button #5,” it means that you have 5 buttons to choose from. Since no repetition, then you would now have 4 buttons to choose from and 3 and so on. 543 2

  11. Permutation Number of arrangements when order MATTERS a, b, c is DIFFERENT than a, c, b r = number of choices n = number of items

  12. CHECK EACH OTHER! …in a nice way, of course  Please swap notes to check if your neighbor wrote down the following correct! n r = number of choices n = number of items

  13. Your foldable… n Factorial • N Factorial For any positive integer n, n! = n(n-1)(n-2)… 3.2.1 Number of Permutations n n = items r = items at a time

  14. Inside… 5! = 5x4x3x2x1 = 120 5!3! = (5x4x3x2x1)(3x2x1) = 120x6 = 720 = = = 120 10 =

  15. From the HW worksheet… #7 The ski club with ten members is to choose three officers captain, co-captain & secretary, how many ways can those offices be filled? Since there are TEN members to choose from and THREE officers to select, 10 = = = We can of multiplication and division = ways to select 3 members

  16. HW examples continued #11 In the Long Beach Air Race six planes are entered and there are no ties, in how many ways can the first three finishers come in? 6 = = = = ways that the first three finishers can come in

  17. HW Assignment • From this same worksheet, • ## 1 – 3 and 7 – 13 only • DO NOT forget that you need to have finished your graphs printed from online by Wednesday. This means that your calculations for your all equations need to be 100% correct. I will be available after school tomorrow for assistance.

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