GPS AlgebraDay 2 (8-16-11)

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.

ProbabilityProbability

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?

Sandwich(2) Side Item(2) Drink(3) Outcomes

chicken

tuna

There are 12 possible lunches.

chips

fruit

chips

fruit

apple juice orange juice milkapple juice orange juice milk

apple juice orange juice milkapple juice orange juice milk

chicken, chips, apple chicken, chips, orange chicken, chips, milkchicken, fruit, apple chicken, fruit, orange chicken, fruit, milk

tuna, chips, apple tuna, chips, orange tuna, chips, milktuna, fruit, apple tuna, fruit, orange tuna, fruit, milk

Easier WayEasier Way

No need to make the tree diagram.

Multiply each of the number of choices(2 sandwiches, 2 sides, 3 drinks)2 2 3

Counting PrincipleCounting 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 PrincipleCounting 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.

1st Song 2nd 3rd 4th 5th 6th 7th 8th Outcomes

There are 40,320 possible song orders.

In this situation it makes more sense to use the Fundamental Counting Principle.

8The 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.

• 7 •6•5 •4 •3•2 •1 =40,320

FactorialFactorial

EXAMPLE with Songs ‘eight factorial’

The product of counting numbers beginning at n and counting backward to 1 is written n! and it’s called n factorial.

8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320

FactorialSimplify each expression.

a. 4!

b. 6!

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.

4 • 3 • 2 • 1 = 24

6 • 5 • 4 • 3 • 2 • 1 = 720

= 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.

President(15) Vice(14) Secretary (13) Treasurer(12) Outcomes

There are 32,760 permutations for choosing the class officers.

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 Secretary TreasurerIs Michael Kim Jane George

the same as…

Jane Michael George Kim ?

Permutation

Notation

PermutationsORDER MATTERS!

Placement Examples: assigned seats, winning a race or running a race, 1st place, 2nd place, etc

PositionsExamples: Pres., Vice Pres, Sec, Tres.

Specific job/choreExamples: Hand out markers, pass out papers, etc

PermutatioPermutationn

When deciding who goes 1st, 2nd, etc., order is important.

*Note  if  n = r   then   nPr  =  n!

A permutation  is an arrangement or listing of objects in a specific order. The order of the arrangement is very important!! 

The notation for a permutation:       nPr = n  is the total number of objects r is the number of objects selected (wanted)

!

( )!

n

n r

PermutatioPermutationsns

Simplify each expression.

a. 12P2

b. 10P4

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?

12 • 11 = 132

10 • 9 • 8 • 7 = 5,040

= 20P4 = 20 • 19 • 18 • 17 = 116,280

Classwork

Practice Workbook Page 341 # – 1-7

Homework

Text book: p. 345 #1 – 4, 20 – 24

Review Worksheet