12­7_Permuations and Combinations 2010 web.notebook
12­7 and 12­8
Permutations and Combinations
December 03, 2010
Warm­up
You are making a sandwich and you can choose from 2 kinds of bread (white or wheat), 3 kinds of meat (ham, turkey or roast beef), and 2 kinds of cheese (cheddar or swiss).
a) Draw a diagram to represent all the possible choices
b) How many sandwich choices are there?
c) Find the probability of making a ham sandwich on wheat bread.
d) Find the probability of making a sandwich with cheddar cheese on white bread. Objectives: To determine whether a problem is a permutation or combination
To calculate the value of a permutation problem
To calculate the value of a combination problem Dec 2­8:52 AM
Dec 2­8:52 AM
Another kind of counting problem involves finding the number of possible arrangements of the items in a set. Permutations
When the order of the items in a set matters, this arrangement is called a permutation.
Dec 2­1:41 PM
Feb 28­11:03 AM
Permutations
Permutations can involve a special mathematical symbol, !. This symbol is called a FACTORIAL. This is what it means:
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
The arrangement of any number of items in a definite order is called a permutation.
The symbol for the number of different arragements when n items are arranged r at a time is nPr
From our swimmer example: 8 items (people) are arranged 8 at a time.
Try this example:
A swimming pool has eight lanes. How many ways can eight swimmers be assigned to these lanes for a race?
In math terms: P = 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1= 40,320
8 8
8 items
arranged 8 at a time
There are 8 people, and we are arranging 8 of them
Feb 28­11:56 AM
Feb 28­12:08 PM
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12­7_Permuations and Combinations 2010 web.notebook
December 03, 2010
Try these ones:
Sometimes we only want to arrange a certain amount of the items at a time.
Example: How many ways can you pick a right, center,
and left fielder from a 9 player baseball team?
Here we have 9 items (players) being arranged 3 at a time.
How many different two letter arrangements can be made from the word HEART?
In a race there are 8 people. How many different ways can our 8 racers come in 1st, 2nd, and 3rd.
9 P3 =
9 items
arranged 3 at a time
Feb 28­12:19 PM
Feb 28­1:32 PM
Permutations Review
Sprinting Events in Track
Permutation: The arrangement of any number of items from a group in a definite order.
P = n r
n items in a group
• n tells you what number goes in the first blank.
In order to qualify for the finals in the 100 ­ meter sprint you need to finish in the top 3 of your heat.
Does it matter who comes in 1st, 2nd, or 3rd in the heat?
arranging r of them at a time
• r tells you how many blanks to draw.
Feb 28­1:01 PM
Dec 2­8:28 AM
Figuring out COMBINATIONS
Important Idea !!!!!
When the order of a collection of objects does not matter, it is called a combination
Looking at our track and field example
There are 8 racers in a preliminary 100 ­ meter sprint. In order to make it to the finals the racers need to finish in the top three.
Step 1: Figure out how many arrangements there are when you care about the order.
Step 2: Get rid of the "repeats."
Dec 2­8:31 AM
Dec 2­8:32 AM
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12­7_Permuations and Combinations 2010 web.notebook
COMBINATION: A collection of objects without regard to order
nCr =
nPr
r!
December 03, 2010
Example
In a smoothie shop you get to pick 4 "extras" to add to the basic smoothie. You can choose from 8 "extras." How many different smoothies can you make?
How many arrangements there are with regard to order
(
(Ask yourself, Does it matter what order I add the extras into my smoothie?
Step 1: Figure out how many arrangements there are when you care about the order.
Gets rid of the "repeats."
Step 2: Get rid of the "repeats."
Dec 2­8:34 AM
Example
10 people report for jury duty. How many different 5­person juries can be chosen?
Dec 2­8:34 AM
Find the combinations:
1. 8C5 =
2. 4C2 =
3. 10C4 =
Dec 2­8:35 AM
When doing problems like these, ask yourself, "DOES ORDER MATTER?"
Dec 2­8:35 AM
Which one is it, permutation or combination?
• Four books pulled at random from a shelf.
• If order matters, the problem deals with permutations.
• Twelve books arranged on a shelf.
• If order does NOT matter, the problem deals with combinations.
• Selecting three bags of chips from a variety pack.
Dec 2­8:36 AM
Dec 2­8:36 AM
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12­7_Permuations and Combinations 2010 web.notebook
December 03, 2010
Homework
Worksheet:
12.7 and 12.8
Feb 28­2:11 PM
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