Counting Principles, Permutations

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Arrangements

• Counting Principles, Permutations Combinations

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Learning Goal

• I will be able to solve counting problems using
  the Fundamental Counting Principle, permutations,
  and combinations.

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Counting Principles
Principle 1 If n(A) and n(B) are the numbers of
ways that Events A and B can occur, respectively,
then the number of ways that Event A and then
Event B can occur is n(A and then B) n(A) x
n(B).
Note We multiply when the situations occur one
after the other.
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Counting Principles
Principle 2 If n(A) and n(B) are the numbers of
ways that Events A and B can occur, respectively,
then the number of ways that Event A or Event B
can occur is n(A or B) n(A) n(B).
Note We add when the situations occur
separately. We assume they cannot both occur.
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Practice

• Ward Robe has 15 pairs of slacks and 23 shirts.
  In how many different ways could he select a
  slacks-and-shirt outfit?

and means multiply 15(23) 345 outfits
2. Natalie Attired has 20 dresses and 17 pants
outfits. In how many different ways could she
select a dress or pants outfit to wear?
or means add 20 17 37 outfits
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Arrangements
Suppose you have four candles you wish to arrange
in a row on a shelf. The four candles are
vanilla, mulberry, orange and raspberry
fragrances. In how many ways can these candles
be arranged?
Strategy Make a List V, M, O, R V, M, R, O V, O,
M, R V, O, R, M V, R, O, M V, R, M, O
M, V, O, R M, V, R, O M, O, R, V M, O, V, R M, R,
O, V M, R, V, O
O, V, M, R O, V, R, M O, M, V, R O, M, R, V O, R,
V, M O, R, M, V
R, M, O, V R, M, V, O R, V, O, M R, V, M, O R, O,
M, V R, O, V, M
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Hey, isnt that the same as 4! ?
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Arrangements
You need to study, practice football, fix dinner,
phone a friend, and go buy a notebook. How many
ways can you arrange your schedule?
Strategy Fill in the blanks
5
4
3
2
1
_______ things can do 1st
_______ things can do 2nd
_______ things can do 3rd
_______ things can do 4th
_______ things can do 5th
Think of this as a do 1st and then do 2nd and
then situation. So, multiply numbers together.
Hmm, that is the same as 5! .
120
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Conclusion
There are n! ways to arrange n items.
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Selected Arrangements
How many ways can you arrange 3 records from a
collection of ten?
Strategy Fill in the Blanks
10
9
8
Multiply!
______ 1st
______ 2nd
______ 3rd
720
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Selected Arrangements
How many ways can you arrange only four of the
Seven Dwarfs in a line?
Multiply!
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6
5
4
_______ 1st
_______ 2nd
_______ 3rd
_______ 4th
840
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Permutation
Permutation is another name for an ordered
arrangement of r items from a set of n. It is
commonly written nPr. ORDER MATTERS! We calculate
it
Yes, that can be found in your calculator. Look
in the MATH PRB menu.
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Permutations
In how many ways can you order the letters in
ADAM?
Strategy Make a List ADAM ADMA AMDA AMAD AADM AAM
D
DAAM DAMA DMAA MADA MAAD MDAA
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Permutations
In how many ways can you order the letters in
MAMA?
Strategy Make a List MAMA MAAM AMMA AMAM AAMM MMA
A
6
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Permutations with identical elements
Sometimes, our set of items contains virtually
identical elements. In this case, we have
virtually identical permutations. We need to
divide by the permutations of the repeats. On the
problem with ADAM, we can arrange four letters in
4! ways, but since A repeats twice, we need to
divide by 2!. So we have 4!/2! 24/2 12. On
the problem with MAMA, we can arrange four
letters in 4! ways, but since A repeats twice and
M repeats twice, we need to divide by 2! twice.
So we have 4!/(2!2!) 24/4 6.
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Class Offices
In how many ways can you fill positions of
President, Treasurer and Historian from our
class? In how many ways can you choose a
governing board of three from our class?
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Combinations
Combination is an unordered arrangement of r
items from a set of n. It is commonly written
nCr. Order does not matter. We calculate it
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Selected Arrangements
How many ways can you pick 4 of the Seven Dwarfs?
35
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Learning Goal

• I will be able to solve counting problems using
  the Fundamental Counting Principle, permutations,
  and combinations.

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End of notes.

• Now do this Gateway Problem
• In the Texas Lotto, players choose six distinct
  numbers between 1 and 54. In how many ways can
  you select the six numbers?

A Gateway Problem is an opportunity to see if you
have mastered the learning goal. You should not
discuss the gateway problem with anyone while
doing it we need to see what you know.
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Gateway Problem Answer25,827,165

• If you got the correct answer, your homework is
  p. 609 Vocab Check 1 5, and 45, 46, 53, 55,
  57, 59, 70, 71, 73.
• If you didnt get the correct answer, your
  homework is p. 609 Vocab Check 1 5, and 39,
  41, 45, 46, 53, 55, 57, 59, 65, 70, 71, 73.