Sherlock Holmes Investigates Door Code with Factorials
"Discover the intriguing world of factorials while joining Sherlock Holmes in cracking door codes! Explore the mathematical concept of factorials and combinations as Sherlock solves mysteries using numbers as clues."
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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?
1! = 1 2! = 2x1= 2 3! = 3x2x1 = 6 4! = 4x3x2x1 = 24 5! = 5x4x3x2x1 = 120 You simply multiply the numbers of whichever n you have.
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?
1 2 3 4 5 5 ways!
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
How many ways could Sherlock open the door if two buttons will unlock it? 20 ways! 5 4 How many possible choices? 5 Now, how many possible choices? 4
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) 1,2,3 1,2,4 1,2,5 1,3,2 There has to be another way
How many different ways could Sherlock open the door if three buttons will unlock it? 5 4 3 A grand total of 60 ways to unlock that door
This does mean that if you now need to use FOUR buttons, it will be: 5 4 3 2 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.
Permutation: -______________________________________________________ Number of arrangements when order MATTERS ______________________________________________________ a, b, c is DIFFERENT than a, c, b r = number of choices -How?_________________________________________________ n = number of items ?! - nPr : ? ? !
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
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 ? ? !
5! = 5x4x3x2x1 = 120 5!3! = (5x4x3x2x1)(3x2x1) = 120x6 = 720 10! 7!3! = 10 9 8 7 6 5 4 3 2 1 120 7 6 5 4 3 2 1 3 2 1 = 10 9 8 3 2 1 = 10! 10 4 ! 10! 6! =10 9 8 7 10?4= =
#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?3= 10 3 ! 7 ! = 10 9 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1 10! = 10! We can of multiplication and division = 10 9 8 = 720 ways to select 3 members
#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 5 4 3 2 1 3 2 1 6! 6! 3 ! 6?3= = 6 3 ! = 6 5 4 = 120 ways that the first three finishers can come in
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.
