Counting Principle and Permutations 10.1 Counting Principle and Permutations

Using a tree diagram to count the ways to perform a task Example: A sporting goods store offers 3 types of snowboards (all-mountain, freestyle, and carving) and 2 types of boots (soft and hybrid). How many choices does the store offer for snowboarding equipment? In a tree diagram, each choice is shown branching out from the previous choices. 6 choices All-mountain, soft 1 Soft All-mountain Hybrid All-mountain, hybrid 2 Soft Freestyle, soft 3 Freestyle Hybrid Freestyle, hybrid 4 Carving Soft Carving, soft 5 Hybrid Carving, hybrid 6

Example When making a sandwich, you can choose from white or wheat bread, tuna or turkey, and lettuce, pickles or mayo. Make a tree diagram showing how many different sandwiches you could make.

Fundamental Counting Principle Another way to count the number of choices at the sporting goods store without making a tree diagram.

Fundamental Counting Principle Two events If one event can occur in m ways and another event can occur in n ways, then the number of ways both events can occur is Three or more events The fundamental counting principle can be extended to three or more events. For example, if three events can occur in m, n and p ways, then the number of ways that all three events can occur is

Example - Framing a picture Picture frames are available in 12 different styles. Each style is available in 55 different colors. You also want blue mat board, which is available is 11 different shades of blue. How many different ways could you frame the picture? 12 55 11 7260 different ways to frame the picture

Counting principle with repetition Standard configuration of a license plate is 1 letter followed by 2 digits followed by 3 letters. a) How many different license plates are possible if letters and digits can be repeated? ___ ___ ___ ___ ___ ___ 26 • 10 • 10 • 26 • 26 • 26 = 45,697,600 plates are possible with repetition

Counting principle with repetition Standard configuration of a license plate is 1 letter followed by 2 digits followed by 3 letters. b) How many different license plates are possible if letters and digits can NOT be repeated? ___ ___ ___ ___ ___ ___ 26 • 10 • 9 • 25 • 24 • 23 = 32,292,000 plates are possible without repetition

For instance, there are 6 permutations of An ordering of n objects is a permutation of the objects. For instance, there are 6 permutations of the letters A, B and C ABC ACB BAC BCA CAB CBA

The number of permutations of n Factorial The symbol for factorial is ! n! is defined where n is a positive integer as follows: n!=n•(n-1)•(n-2)•…•3•2•1 Example: 5!=5•4•3•2•1 The number of permutations of n distinct objects is n!

Find the number of permutations You are creating a playlist of 10 songs. How many different ways could you arrange the songs? 10! = 10•9•8•7•6•5•4•3•2•1=3,628,800 ways to arrange the songs

What if? What if I wanted to burn just 4 of my 10 songs onto a demo CD for my band – how many ways could I arrange the 4 songs? Time for a new formula…

Permutation of n objects taken r at a time The number of permutations of r objects taken from a group of n distinct objects is given by this formula Order matters!

Back to our demo CD... What if I wanted to burn just 4 of my 10 songs onto a demo CD for my band – how many ways could I arrange the 4 songs? I have 10 songs and I want to arrange 4 of them on a CD, so this is a permutation where n=10 and r=4.

Permutations with repetition The number of distinguishable permutations of n objects where one object is repeated time and another object is repeated times, and so on, is:

Example Find the number of distinguishable permutations of the letters in MIAMI. There are 5 letters is MIAMI. M and I are each repeated 2 times. The number of distinct permutations is