Transcript Counting Principles and Permutations

Homework
Review
GPS Algebra
UNIT QUESTION: How do you use
probability to make plans and predict
for the future?
Standard: MM1D1, MM1D2, MM1D3
Today’s Question:
How can I find the different ways to
arrange the letters in the word
“PENCIL”?
Standard: MM1D1.b.
Probability
The Counting Principle &
Permutations
Use an appropriate method to find the number of
outcomes in each of the following situations:
1. Your school cafeteria offers chicken or tuna sandwiches; chips
or fruit; and milk, apple juice, or orange juice. If you purchase
one sandwich, one side item and one drink, how many different
lunches can you choose? There are 12 possible lunches.
Sandwich(2)
Side Item(2)
chips
chicken
fruit
chips
tuna
fruit
Drink(3)
Outcomes
apple juice
orange juice
milk
apple juice
orange juice
milk
chicken, chips, apple
chicken, chips, orange
chicken, chips, milk
chicken, fruit, apple
chicken, fruit, orange
chicken, fruit, milk
apple juice
orange juice
milk
apple juice
orange juice
milk
tuna, chips, apple
tuna, chips, orange
tuna, chips, milk
tuna, fruit, apple
tuna, fruit, orange
tuna, fruit, milk
Easier Way
No need to make the tree diagram.
Multiply each of the number of choices
(2 sandwiches, 2 sides, 3 drinks)
223
Counting Principle
• At a sporting goods store, skateboards are
available in 8 different deck designs.
Each deck design is available with 4
different wheel assemblies. How many
skateboard choices does the store offer?
32
Counting Principle
• A father takes his son, Marcus, to
Wendy’s for lunch. He tells Marcus he
can get a 5 piece nuggets, a spicy
chicken sandwich, or a single for the
main entrée. For sides, he can get
fries, a side salad, potato, or chili. And
for drinks, he can get a frosty, coke,
sprite, or an orange drink. How many
options for meals does Marcus have?
48
Many iPods can vary the order in which songs are
played. Your mp3 currently only contains 8 songs. Find
the number of orders in which the songs can be played.
There are 40,320 possible song orders.
1st Song
2nd
3rd
4th
5th
6th
7th
8th
Outcomes
In this situation it makes more sense to use the
Fundamental Counting Principle.
8 • 7 • 6 • 5 • 4 • 3 • 2 •1 = 40,320
The solutions in examples 3 and 4 involve the
product of all the integers from n to one. The
product of all positive integers less than or equal
to a number is a factorial.
Factorial
The product of counting numbers
beginning at n and counting
backward to 1 is written n! and it’s
called n factorial.
EXAMPLE with Songs ‘eight factorial’
8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320
Factorial
Simplify each expression.
a. 4!
4 • 3 • 2 • 1 = 24
b. 6!
6 • 5 • 4 • 3 • 2 • 1 = 720
c. For the 8th grade field events there are
five teams: Red, Orange, Blue, Green,
and Yellow. Each team chooses a
runner for lanes one through 5. Find the
number of ways to arrange the runners.
= 5! = 5 • 4 • 3 • 2 • 1 = 120
The student council of 15 members must choose a
president, a vice president, a secretary, and a treasurer.
There are 32,760 permutations for choosing the class officers.
President(15) Vice(14)
Outcomes
Secretary (13)
Treasurer(12)
In this situation it makes more sense to use the
Fundamental Counting Principle.
15•14 • 13 •12 =
32,760
Let’s say the student council members’ names were:
John, Miranda, Michael, Kim, Pam, Jane, George,
Michelle, Sandra, Lisa, Patrick, Randy, Nicole, Jennifer,
and Paul. If Michael, Kim, Jane, and George are elected,
would the order in which they are chosen matter?
President
Vice President
Is Michael
Kim
Secretary
Jane
Treasurer
George
the same as…
Jane
Michael
George
Kim ?
Permutation
Notation
Permutations
ORDER MATTERS!
Placement
Examples: assigned seats, winning a race or
running a race, 1st place, 2nd place, etc
Positions
Examples: Pres., Vice Pres, Sec, Tres.
Specific job/chore
Examples: Hand out markers, pass out
papers, etc
Permutation
When deciding who goes 1st, 2nd, etc., order is
important.
A permutation is an arrangement or
listing of objects in a specific order.
The order of the arrangement is very important!!
P
n!
(n  r )!
The notation for a permutation:
n r=
n is the total number of objects
r is the number of objects selected (wanted)
*Note if n = r then nPr = n!
Permutations
Simplify each expression.
a.
12P2
b. 10P4
12 • 11 = 132
10 • 9 • 8 • 7 = 5,040
c. At a school science fair, ribbons are
given for first, second, third, and fourth
place, There are 20 exhibits in the fair.
How many different arrangements of
four winning exhibits are possible?
= 20P4 = 20 • 19 • 18 • 17 = 116,280
Classwork
Practice Workbook Page 360 #12 – 20
Homework
Text book: p. 344 #1 – 6, 25 – 28
Review Worksheet