n Factorial Permutations What is n Factorial The

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! = 2 x 1= 2 � 3! = 3 x 2 x 1 = 6 � 4! = 4 x 3 x 2 x 1 = 24 � 5! = 5 x 4 x 3 x 2 x 1 = 120 You simply multiply the numbers of whichever “n” you have.

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?

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

(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 3, 1 3, 2 3, 4 5, 1 5, 2 5, 3 1, 5 2, 1 3, 5 4, 1 5, 4 2, 3 2, 4 2, 5 4, 2 4, 3 4, 5

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

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, 3 1, 2, 4 1, 2, 5 1, 3, 2 … There has to be another way…

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

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.

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

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

Your foldable… N Factorial For any positive integer n, n! = n(n-1)(n-2)… 3. 2. 1 Number of Permutations n Factorial

Inside…

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? We can of multiplication and division

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?

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.