Warm-Up 4/29 simplify1.

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Rigor:You will learn how to find the number of

possible outcomes using the Fundamental Counting Principle, permutations and

combinations.

Relevance:You will be able to solve probability

problems using the Fundamental Counting Principle, permutations and combinations.

0-4 Counting Techniques

Probability is a measure of the chance that a given event will occur.

Outcome is the result of a probability experiment or an event.

Sample Space is the set of all possible outcomes of an experiment

A tree diagram can be used to list all outcomes in a sample space.

EX: Blood type R H factor

A

B

AB

O

+– +– +–

+–

Fundamental Counting Principle

Words – The number of possible outcomes in a sample space can be determined by multiplying the number of possible outcomes for each event.

Symbols – 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 N can occur in m•n ways.

This rule can be extended for 3 or more events.

4 possible blood types and 2 possible RH factors then there are 42 or 8 possible ways to label blood.

Example 1: A bicycle manufacturer makes five- and ten-speed bikes in seven different colors and four different sizes. How many different bikes do they make?

& 4 frame choices7 color choices, 2 gear choices,

2 ∙7 ∙ 4=56

56 different bicycles can be made.

When there are m ways to do one thing, and n ways to do another, then there are m•n ways of doing both.

Your Turn:You are picking out an outfit.

There are 3 pants and 2 shirts.

How many possible outfits? (draw a tree diagram)

3•2 = 6 outfits1 2

3

ab

ab

ab

Permutation is an arrangement of a group of distinct objects in a certain ORDER.

There are 6 permutations of the letters A, B and C.

ABC, ACB, BAC, BCA, CAB, CBA

_______ _________ ________ = choices for first letter

choices for second letter

choices for third letter

total number of permutations

3•2•1 = 6 permutations

3•2•1 = 3!

Factorial

The number of permutations of distinct objects is !.0 !=1

Two types of permutation:

No Repetition: for example the first three people in a running race. You can't be first and second.

The number of available choices reduces each time.

n × (n – 1) × (n – 2) × …

Repetition is Allowed: such as the combination of a lock. It could be "333". These are the easiest to calculate!n × n × ... × n, (r times) = nr

Example 2a: There are 8 finalists in a band competition. How many different ways can the bands be ranked if they cannot receive the same ranking?

8 !

The bands can be ranked in 40,320 different ways.

8 ∙7 ∙ 6 ∙5 ∙ 4 ∙ 3 ∙2 ∙ 1=40,320

Example 2b: How many different ways are their to order 16 pool balls?

16 !

different ways.

16 ∙ 15 ∙14 ∙ …∙ 3 ∙ 2∙ 1=20,922,789,888,000

What if we don't want to choose them all, just 3 of them?

16 ∙ 15 ∙14=3360

different ways.

Example 2c: How many different ways are their to unlock the lock?

In the lock, there are 10 numbers to choose from (0,1,...9)

There are 1000 permutations to unlock the lock.

10 ∙ 10 ∙10=103=1000

So, we should really call this a "Permutation Lock"!

There are 3 numbers.

Your Turn:1. You have 5 books to arrange on a shelf. How many

different ways can this be done?

2. To create an entry code for a push-button lock, you need to first choose a letter and then, three single-digit numbers. How many different entry code can you create?

3. In how many ways can 25 runners finish first, second and third?

5 !5 ∙4 ∙ 3 ∙ 2 ∙1=120

26 ∙ 10 ∙10 ∙ 10=26000

25 ∙ 24 ∙ 23=13800

Key Concept: PERMUTATIONS of n Objects Taken r at a Time:

The number of permutations of r objects taken from a group of n distinct objects is given by

Going back to the pool ball question. When we only want 3 out of the 16 pool balls.

Example 3: How many ways can two students be assigned to five tutors if only one student is assigned to each tutor?

5 different tutors taken 2 at a time.

The students can be assigned in 20 different ways.

𝑛𝑃𝑟=𝑛 !

(𝑛−𝑟 ) !

n = 5

r = 2 5𝑃2=5 !

(5−2 )!

¿5 ∙4 ∙ 3 ∙ 2∙ 1

3 ∙ 2∙ 1

¿2 0

Your Turn:A photographer has matted and framed 15 photographs and needs to select 10 for a show. How many ways can the photographs be arranged?

𝑛𝑃𝑟=𝑛 !

(𝑛−𝑟 ) !¿15 ∙ 14 ∙ 13 ∙12 ∙11 ∙ 10 ∙ 9 ∙ 8 ∙7 ∙6

¿10,897,286,400

In English we use the word "combination" loosely

• My fruit salad is a combination of apples, grapes and bananas

• The combination to the safe was 472

– We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

– Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2

Star This

If the order doesn't matter, it is a Combination.

If the order does matter it is a Permutation.

To help you to remember, think “Combination”… Committee “Permutation” ... Position

A combinations is a selection of distinct objects in which the order of the objects selected is not important.

Key Concept: COMBINATIONS of n Objects Taken r at a Time:

The number of combinations of r objects taken from a group of n distinct objects is given by

Example 4: How many ways are there to chose 5 cards from a standard deck of 52 playing cards?

52 cards taken 5 at a time and order doesn’t matter.

There are 2,598,960 ways to choose 5 cards from a standard deck of playing cards.

𝑛𝐶𝑟=𝑛 !

(𝑛−𝑟 ) !𝑟 !

n = 52

r = 5 52𝐶5=52!

(52 −5 ) !5 !

¿52∙ 51 ∙50 ∙ 49 ∙ 48 ∙ 47 !

47 ! ∙5 ∙ 4 ∙3 ∙2 ∙1¿2 ,598,960

¿52 !

4 7 !5 ! 210

Example 5: Twenty-five students write their name on paper. Then three names were picked at random to receive prizes. (permutation or combination)

a. Selecting 3 people to each receive a “no homework” pass.

b. Selecting 3 students to receive one of the following prizes: 1st prize – a new graphing calculator; 2nd prize- a “no homework” pass; 3rd prize – a new pencil.

Order doesn’t matter so this is a combination.

Order does matter so this is a permutation.

Your Turn:A restaurant offers a total of 8 side dishes. How many different ways can a customer choose 3 side dishes?

¿56𝑛𝐶𝑟=𝑛 !

(𝑛−𝑟 ) !𝑟 !

AssignmentProb/Stats #1 WS, 1-19 All Conics Project Due Dates:Sections 3 & 4 + 1 & 2 due tomorrowYOU MUST HIGHLIGH CORRECTIONS USING A BLUE HIGHLIGHTER.

Warm-Up 4/29

1. A folders come in 8 different colors. How many different ways can I choose 4 colors?

2. A push-button lock, has a code that you need to first choose a letter and then, 4 single-digit numbers. How many different entry code can you create?

¿708𝐶4=8 !

(8 − 4 ) !4 ! ¿8 ∙7 ∙6 ∙5 ∙ 4 !4 ! ∙ 4 ∙3 ∙ 2∙ 1

2

26 ∙ 10 ∙10 ∙ 10 ∙ 10¿260 ,0 00

Conics Project Due Dates:Sections 3 & 4 + 1 & 2 due tomorrowYOU MUST HIGHLIGH CORRECTIONS USING A BLUE HIGHLIGHTER.