course: math lit. aim: counting principle aim: how do i count the ways? do now: use , , or both to...

$: Course: Math Lit. Aim: Counting Principle Aim: How do I count the ways? Do Now: Use , , or both to make the following statement true. {s, r, t} _____$
Course: Math Lit.Aim: Counting Principle

Aim: How do I count the ways?

Do Now:

Use , , or both to make the following statement true.

{s, r, t} _____ {s, r, t}


Combinatorics

Combinatorics – the study of counting the different outcomes of some task.

Ex. A coin is flipped. Two possible outcomes: set {H, T}

Ex. A die is tossed. 6 possible outcomes: {1, 2, 3, 4, 5, 6}

List and then count the number of different outcomes that are possible when one letter from the word Tennessee is chosen. {T, e, n, s}

List and then count the number of different outcomes that are possible when one letter from the word Mississippi is chosen. {M, i, s, p}

Counting by Forming a List


Combinatorics

Experiment – An activity with an observable outcome.

Sample Space – set of possible outcome for an experiment.

Event – one or more of the possible outcomes of an experiment. An event is a subset of the sample space.

Ex. Flipping a coin resulting in H; rolling a 5 when a die is tossed. Choosing the

letter T are all experiments and a subset of each respective sample space.

Each of these experiments are single or simple experiments – a single outcome.


Model Problem

One number is chosen from the sample space S = {1, 2, 3, 4, 5, 6, 7, 8 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

List the elements in the following events.

a. The number is even

b. The number is divisible by 5.

c. The number is prime.

{2, 4, 6, 8, 10, 12, 14, 16, 18, 20}

{5, 10, 15, 20}

{2, 3, 5, 7, 11, 13, 17, 19


Counting by Making a Table

Multi-stage or compound experiments – experiments with more than one stage.

Ex. rolling two dice: one red, one green.

36 possible outcomes

How many red-green dice tosses results in a sum of seven? 6


Model Problem

Two-digits numbers are formed from the digits 1, 3, and 8. Find the sample space and determine the number of elements in the sample space.

1 3 8

1

3

8

11 13 18

31 33 38

81 83 88

{11, 13, 18, 31, 33, 38, 81, 83, 88}


Tree Diagram

Tree diagram – a method for organizing multi-staged or compound experiment.

Ex. The options for today’s lunch are the following:

Main Course: spaghetti, hamburger or hot dog

Drink: milk or coke

Dessert: ice cream, apple pie or chocolate cake

Multi-stage experiment


Spaghetti

Hamburger

Hotdog

Main Course Drink

Milk

Coke

Dessert

Ice CreamApple PieChocolate Cake


S M IS M AS M C

S C IS C AS C C





Milk

Coke

Milk

Coke

H M A

Ht M IHt M AHt M C

H M I

H M C

H C IH C AH C C

Ht C IHt C AHt C C

Sample Space

Tree Diagram


Model Problem

A true/false test consists of 10 questions. Draw a tree diagram to show the number of ways to answer the first three questions.

T

F

T

F

T

F

T

F

T

F

T

F

T

F

TTT

TTF

TFT

TFF

TFF

FTF

FFT

FFF


MJ Petrides

Outerbridge Crossing

Great Adventure

How many different ways will get us from MJ Petrides to Great Adventure?

Tracing the different routes we find there are 6 different routes.

Is there a shortcut method for finding how many different routes there are?

3

2

Fundamental Counting Principle

3 x 2 = 6


To find the total number of possible outcomes in a sample space, multiply the number of choices for each stage or event...

in other words...

If event M can occur in m ways and is followed by event N that can occur in n ways, then the event M followed by event N can occur in m · n ways.

Main Idea

Counting Principle 2 events: m · n 3 events: m · n · o 4 events: m · n · o · p 5 events: etc.

Fundamental Counting Principle


18x x =

Spaghetti

Hamburger

Hotdog

Main Course Drink

Milk

Coke

DessertIce CreamApple PieChocolate Cake


S M IS M AS M C

S C IS C AS C C





Milk

Coke

Milk

Coke

H M A

Ht M IHt M AHt M C

H M I

H M C

H C IH C AH C C

Ht C IHt C AHt C C

Sample Space

3 2 3


Jamie has 3 skirts - 1 blue, 1 yellow, and 1 red. She has 4 blouses - 1 yellow, 1 white, 1 tan and 1 striped. How many skirt-blouse outfits can she choose?

Blue

Yellow

Red

yellow

striped

white

tan

yellow

striped

white

tan

yellow

striped

white

tan

Skirt blouse3 4 12 outcomes in sample

space

B Y

B W

B T

B S

Y Y

Y W

Y T

Y S

R Y

R W

R T

R S

Counting Principle 2 events: m · n 3 · 4 = 12


Model Problem

In horse racing, a trifecta consists of choosing the exact order of the first three horses across the finish line. If there are eight horses in a race, how many trifectas are possible, assuming no ties.

1st place 2nd place 3rd place

8 7 6x x = 336

Nine runners are entered in a 100-meter dash for which a gold, silver, and bronze medal will be awarded for 1st, 2nd and 3rd place finishes. In how many ways can the medals be awarded? 504


Counting with and without Replacement

From the letters a, b, c, d, and e, how many four letter groups can be formed if

a. a letter can be used more than once?

b. each letter can be used exactly once?

1st 2nd 3rd 4th

5 5 5 5. . . = 54 = 625

1st 2nd 3rd 4th

5 4 3 2. . . = 120


Model Problem

A four-digit serial number is to be created from the digits 0 through 9. How many of these serial numbers can be created if 0 can not be the first digit, no digit may be repeated, and the last digit must be 5?

1) 448 2) 2240 3) 504 4) 2,520

possible outcomes

E1 E2 E3 E4

8 8 7 1 = 448

0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10 possible outcomes to start


Determine the number of outcomes:

4 coins are tossed

A die is rolled and a coin is tossed

A tennis club has 15 members: 8 women and seven men. How many different teams may be formed consisting of one woman and one man on each team?

A state issues license plates consisting of letters and numbers. There are 26 letters, and the letters may be repeated on a plate; there are 10 digits, and the digits may be repeated. The how many possible license plates the state may issue when a license consists of: 2 letters, followed by 3 numbers, 2 numbers followed by 3 letters.

course: math lit. aim: counting principle aim: how do i count the ways? do now: use , , or both to...

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