Course: Math Lit. Aim: Counting Principle Aim: How do I count the ways? Do Now: Use , , or both to...
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Transcript of 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} _____ {s, r, t}
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
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.
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
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}
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
Spaghetti
Hamburger
Hotdog
Main Course Drink
Milk
Coke
Dessert
Ice CreamApple PieChocolate Cake
Ice CreamApple PieChocolate Cake
S M IS M AS M C
S C IS C AS C C
Ice CreamApple PieChocolate Cake
Ice CreamApple PieChocolate Cake
Ice CreamApple PieChocolate Cake
Ice CreamApple PieChocolate Cake
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
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
18x x =
Spaghetti
Hamburger
Hotdog
Main Course Drink
Milk
Coke
DessertIce CreamApple PieChocolate Cake
Ice CreamApple PieChocolate Cake
S M IS M AS M C
S C IS C AS C C
Ice CreamApple PieChocolate Cake
Ice CreamApple PieChocolate Cake
Ice CreamApple PieChocolate Cake
Ice CreamApple PieChocolate Cake
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
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
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
Course: Math Lit.Aim: Counting Principle
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.