12.5 permutations 1
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Transcript of 12.5 permutations 1
Lesson 12.5, For use with pages 675-679
1. Choose one of 4 friends, one of 4 restaurants, and one of 3 meeting times.
Use the counting principle to find the number of possible choices.
You NEED a scientific calculator TODAY!
2. How many seven digit phone numbers are possible if the first digit cannot be a 0?
Lesson 12.5, For use with pages 675-679
1. Choose one of 4 friends, one of 4 restaurants, and one of 3 meeting times.
ANSWER 48
Use the counting principle to find the number of possible choices.
You NEED a scientific calculator TODAY!
2. How many seven digit phone numbers are possible if the first digit cannot be a 0?
ANSWER 9 x 10 x 10 x 10 x 10 x 10 x 10 = 9,000,000
Permutations(Arrangements)
Section 12.5
P. 675
Essential Questions
• What are the differences between permutations and combinations?
• What are the differences between odds and probability?
• How is probability used to make predictions?
• What are the differences between experimental and theoretical probabilities?
• When looking at options or choices, we often look at the arrangement, or order, of people, letters, numbers, etc.
A permutation is an arrangement in which ORDER is important.
Arranging “ABC” is different than “ACB”
• Let’s look at the three letters – A, B, C
• How many different ways can I arrange or order these letters?
• What if I add “D” to the list? Now, how many different arrangements?
• Let’s look at the three letters – A, B, C
• How many different ways can I arrange or order these letters?
• What if I add “D” to the list? Now, how many different arrangements?
6 ways
24 ways
EXAMPLE 1 Counting Permutations
Music
You have five CDs. You can use the counting principle to count the number of permutations of 5 CDs. This is the number of different orders in which you can listen to the CDs.
Choices for 1st CD
Choices for 2nd CD
Choices for 3rd CD
Choices for 4th CD
Choices for 5th CD
5 4 3 2 1
ANSWER
You can listen to the CDs in 120 different orders.
= 120
Swimming Event• There are 8 lanes for the swimming event. If
the top 2 swimmers are assigned to lanes 3, and 4 in how many ways can the other 6 swimmers be assigned to their lanes?
• (Hint: -- you are only looking at 6 lanes)• For the first of the remaining 6 swimmers, there are 6
possible lanes assignments. Once a selection has been made, there are only 5 lanes left from which to assign the next swimmer. After the second swimmer is assigned, there are only 4 lanes left, and so on.
• 6 x 5 x 4 x 3 x 2 x 1 = 720 ways
• Factorial Notation is used to write the product when the factors are consecutive whole numbers.
• 4 factorial is written as 4! and it means
4 x 3 x 2 x 1 or 24
Find the factorial key on your calculator.
GUIDED PRACTICE for Examples 1 and 2
1. Evaluate 3!.
3! = 3 2 1
2. Evaluate 6!.
6! = 6 5 4 3 2 1
3. Evaluate 1!.
1!
= 720
= 6
= 1
• How many ways can the letters in these words be arranged?
MATH PROBABILITY
GEOMETRY VOLUME
4! = 24 11! = 39,916,800
8! = 40,320 6! = 720
Permutations
n = total to select from
r = the number you are selecting
EXAMPLE 3 Evaluating a Permutation
Poetry
Two students are chosen from a group of 6 to read the first and second poems at the school’s poetry reading. To find how many different ways the students can be chosen, find 6P2.
6P2– == 6!
(6 2)!6!4!
Use permutation formula.
= 6 5 4 3 2 14 3 2 1
Divide out common factors.
= 30 Multiply.
ANSWERThe students can be chosen in 30 ways.Using the counting principle we would have found it by taking 6 x 5 = 30
• From the 9 players in the batting order, in how many ways can a baseball manager choose and order
the first 3 batters?• 9 x 8 x 7 = 504 or 9P3 = 504• the first 4 batters?• 3024 ways the first 5 batters? 15,120 ways
• There are 10 candidates for student government. There are 4 different positions. In how many ways can the student government be chosen?
• 10P4 = 5040 ways
• Remember: these examples all involve ORDER – arrangements / permutations
On Line Calculators
• Permutations and Combinations– http://www.calctool.org/CALC/math/
probability/combinations– http://www.mathsisfun.com/combinatorics/
combinations-permutations-calculator.html
• Factorials– http://joemath.com/math124/Calculator/
factorial.htm
• www.lewiscentral.k12.ia.us/shipp
• Assignment: P. 677 #1-21