9.6 Counting Principles Permutations Combinations.

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Presentation transcript:

9.6 Counting Principles Permutations Combinations

Fundamental Counting Principle If two choices happen in different ways, one choice is which pants to wear (called “p” ways), another choice is which shirt to wear (called “s” ways), the number of choices are “p” times “s”. If you wanted to include what type shoes ( called “g”), you would have “p” times “s” times “g” choices

Passwords If a password must contain one number and must have seven letters. How many different choices are there? __ __ __ __

Passwords If a password must contain one number and must have seven letters. How many different choices are there? 26 · 26 · 26 · 26 · 26 · 26 · 26 · 10 = 8.03 X 10 10

Permutations Permutation is the act of changing the arrangement of a given number of elements. A,B,CA,C,B B,A,CB,C,A C,A,BC,B,A

Permutations How many different ways in which to arrange 5 books? 1 st book, we have 5 choices 2 nd book, we have 4 choices 3 rd book, we have 3 choices 4 th book, we have 2 choices 5 th book, we have 1 choice By the Fundamental Counting Principle how many choices? 5 · 4 · 3 · 2 · 1= 5! = 120

Permutations The number of Permutations of n items is n! If there are n = 7 choices of toppings on a pizza, how many pizzas can be made? 7! =5040

Permutation when only wanting a limited number. 8 boxes of cereal but only have shelf space for 3 boxes. 1 st space 8 choices 2 nd space 7 choices 3 rd space 6 choices Fundamental Counting Principle 8 · 7 · 6 = 336 choices

Permutation when only wanting a limited number. Permutations of n elements taken r at a time.

Permutation when only wanting a limited number. Permutations of n elements taken r at a time.

What if you where looking at arranging the letters in race car. How many different permutations. Remember race car is a palindrome (same read from left to right or right to left). There are 7 letters with 3 repeating letters a, c and r.

Distinguishable Permutations race car

Combinations Here order is not important, in other words A,B,C and B,C,A are the same because they have the same letters.

Given 5 boxes, how many combinations can there be of 3 boxes? Box A, B, C, D, E {A,B,C} : {B,C,D} : {C,D,E} {A,B,D} : {B,C,E} {A,B,E} : {B,D,E} {A,C,D} : {A,C,E} :10 combinations {A,D,E} :

Combinations of n elements taken r at a time

Homework Page 669 – 671 #10, 14, 18, 22, 26, 27, 30, 34, 36, 42, 52, 60