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9-6. Combinations. Warm Up. Problem of the Day. Lesson Presentation. Pre-Algebra. 9-6. Permutations and Combinations. Pre-Algebra. Warm Up Find the number of possible outcomes. 1. bagels: plain, egg filling: turkey, ham, roast beef, tuna, chicken, cheese, sourcream - PowerPoint PPT Presentation
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Pre-Algebra
9-6 Permutations and Combinations9-6 Combinations
Pre-Algebra
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Pre-Algebra
9-6 Permutations and Combinations
Warm UpFind the number of possible outcomes.
1. bagels: plain, egg filling: turkey, ham, roast beef, tuna, chicken,
cheese, sourcream
2. eggs: scrambled, over easy, hard boiled, raw meat: sausage patty, sausage link, bacon, ham,
steak, pork
3. How many different 3–digit phone extensions are possible?
Pre-Algebra
9-6 Permutations and Combinations
Pre-Algebra
9-6 Permutations and Combinations
Learn to find combinations.
Pre-Algebra
9-6 Permutations and Combinations
Vocabulary
combination
Pre-Algebra
9-6 Permutations and Combinations
The factorial of a number is the product of all the whole numbers from the number down to 1. The factorial of 0 is defined to be 1.
5!5! = 55 • 44 • 33 • 22 • 11
Read 5! as “five factorial.”
Reading Math
Pre-Algebra
9-6 Permutations and Combinations
A combination is a selection of things in any order.
Pre-Algebra
9-6 Permutations and Combinations
If no letter is used more than once, there is only 1 combination of the first 3 letters of the alphabet. ABC, ACB, BAC, BCA, CAB, and CBA are considered to be the same combination of A, B, and C because the order does not matter.
If no letter is used more than once, there are 10 combinations of the first 5 letters of the alphabet, when taken 3 at a time. To see this, look at the list of permutations on the next slide.
Pre-Algebra
9-6 Permutations and Combinations
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
ACB ADB AEB ADC AEC AED BDC BEC BED CED
BAC BAD BAE CAD CAE DAE CBD CBE DBE DCE
BCA BDA BEA CDA CEA DEA DBC CEB DEB DEC
CAB DAB EAB DAC EAC EAD DCB EBC EBD ECD
CBA DBA EBA DCA ECA EDA DBC ECB EDB EDC
These 6 permutations are all the same combination.
In the list of 60 permutations, each combination is repeated 6 times. The number of combinations is = 10.
6010
Pre-Algebra
9-6 Permutations and Combinations
Pre-Algebra
9-6 Permutations and Combinations
Additional Example 3A: Finding Combinations
Mary wants to join a book club that offers a choice of 10 new books each month.
A. If Mary wants to buy 2 books, find the number of different pairs she can buy.10 possible books
2 books chosen at a time
10!2!(10 – 2)!
= 10!2!8!
= 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1(2 • 1)(8 • 7 • 6 • 5 • 4 • 3 • 2 • 1)
10C2 =
= 45
There are 45 combinations. This means that Mary can buy 45 different pairs of books.
Pre-Algebra
9-6 Permutations and Combinations
Additional Example 3B: Finding Combinations
B. If Mary wants to buy 7 books, find the number of different sets of 7 books she can buy.
10 possible books
7 books chosen at a time
10!7!(10 – 7)!
= 10!7!3!10C7 =
10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 (7 • 6 • 5 • 4 • 3 • 2 • 1)(3 • 2 • 1)
= = 120
There are 120 combinations. This means that Mary can buy 120 different sets of 7 books.
Pre-Algebra
9-6 Permutations and Combinations
Try This: Example 3A
Harry wants to join a DVD club that offers a choice of 12 new DVDs each month.
A. If Harry wants to buy 4 DVDs, find the number of different sets he can buy.
12 possible DVDs
4 DVDs chosen at a time
12!4!(12 – 4)!
= 12!4!8!
= 12 • 11 • 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1(4 • 3 • 2 • 1)(8 • 7 • 6 • 5 • 4 • 3 • 2 • 1)
12C4 =
= 495
Pre-Algebra
9-6 Permutations and Combinations
Try This: Example 3A Continued
There are 495 combinations. This means that Harry can buy 495 different sets of 4 DVDs.
Pre-Algebra
9-6 Permutations and Combinations
Try This: Example 3B
B. If Harry wants to buy 11 DVDs, find the number of different sets of 11 DVDs he can buy.
12 possible DVDs
11 DVDs chosen at a time
12!11!(12 – 11)! =
12!11!1!
= 12 • 11 • 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1(11 • 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1)(1)
12C11 =
= 12
Pre-Algebra
9-6 Permutations and Combinations
Try This: Example 3B Continued
There are 12 combinations. This means that Harry can buy 12 different sets of 11 DVDs.
Pre-Algebra
9-6 Permutations and Combinations
Evaluate each expression.
1. 9!
2.
3. There are 8 hot air balloons in a race. In how many possible orders can all 8 hot air balloons finish the race?
4. A group of 12 people are forming a committee. How many different 4-person committees can be formed?
Lesson Quiz
3024
362,880
40,320
495
9!5!
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