Learn to find permutations and combinations'

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Learn to find permutations and combinations.
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Vocabulary
factorial permutation combination
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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 4 3 2 1
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Additional Example 1A 1B Evaluating
Expressions Containing Factorials
Evaluate each expression.
A. 8!
8 7 6 5 4 3 2 1 40,320
8!
B.
6!
Write out each factorial and simplify.
Multiply remaining factors.
8 7 56
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Additional Example 1C Evaluating Expressions
Containing Factorials
C.
Subtract within parentheses.
10 9 8 720
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Try This Example 1A 1B
Evaluate each expression.
A. 10!
10 9 8 7 6 5 4 3 2 1 3,628,800
7!
B.
5!
Write out each factorial and simplify.
Multiply remaining factors.
7 6 42
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Try This Example 1C
C.
Subtract within parentheses.
9 8 7 504
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A permutation is an arrangement of things in a
certain order.
If no letter can be used more than once, there
are 6 permutations of the first 3 letters of the
alphabet ABC, ACB, BAC, BCA, CAB, and CBA.
3 choices
2 choices
1 choice


The product can be written as a factorial.
3 2 1 3! 6
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If no letter can be used more than once, there
are 60 permutations of the first 5 letters of the
alphabet, when taken 3 at a time ABE, ACD, ACE,
ADB, ADC, ADE, and so on.
5 choices
4 choices
3 choices
?
?
60 permutations
Notice that the product can be written as a
quotient of factorials.
60 5 4 3
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Additional Example 2A Finding Permutations
Jim has 6 different books.
A. Find the number of orders in which the 6 books
can be arranged on a shelf.
6P6
720
There are 720 permutations. This means there are
720 orders in which the 6 books can be arranged
on the shelf.
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Additional Example 2B Finding Permutations
B. If the shelf has room for only 3 of the books,
find the number of ways 3 of the 6 books can be
arranged.
6 5 4
6P3
120
There are 120 permutations. This means that 3 of
the 6 books can be arranged in 120 ways.
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Try This Example 2A
There are 7 soup cans in the pantry.
A. Find the number of orders in which all 7 soup
cans can be arranged on a shelf.
7P7
5040
There are 5040 orders in which to arrange 7 soup
cans.
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Try This Example 2B
There are 7 soup cans in the pantry.
B. If the shelf has only enough room for 4 cans,
find the number of ways 4 of the 7 cans can be
arranged.
7P4
7 6 5 4
840
There are 840 permutations. This means that the 7
cans can be arranged in the 4 spaces in 840 ways.
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A combination is a selection of things in any
order.
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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.
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ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ACB ADB AE
B ADC AEC AED BDC BEC BED CED BAC BAD BAE CAD CAE
DAE CBD CBE DBE DCE BCA BDA BEA CDA CEA DEA DBC CE
B 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.
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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.
10C2

45
There are 45 combinations. This means that Mary
can buy 45 different pairs of books.
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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.
10C7
120
There are 120 combinations. This means that Mary
can buy 120 different sets of 7 books.
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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.
12C4
495
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Try This Example 3A Continued
There are 495 combinations. This means that Harry
can buy 495 different sets of 4 DVDs.
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Try This Example 3B
B. If Harry wants to buy 11 DVDs, find the number
of different sets of 11 DVDs he can buy.
12C11
12
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Try This Example 3B Continued
There are 12 combinations. This means that Harry
can buy 12 different sets of 11 DVDs.
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Lesson Quiz
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?
362,880
3024
40,320
495