Bell Ringer

Samson wants to climb a mountain. He knows that if the slope is over 2, he will fall. Will he fall?

(1, 0)

(5, 9)m = 9 – 0 9 – 5

94

=

m = 2.25Samson will fall!

No! I’m falling!

Homework

1. 282. 103. 354. 205. 1266. 220

7. 728. 16809. 21010. 504011.

95,04012. 156

13. 79214. 1320

Simple Counting Techniques

Combinations (Formula)

(Order doesn’t matter! AB is the same as BA)

nCr =

Where:n = number of things you can choose

fromr = number you are choosing

n!r! (n – r)!

16•15•14•13•12•11•10•9•8•7•6•5•4•3•2•12•1 (14•13•12•11•10•9•8•7•6•5•4•3•2•1)

Combinations

The summer Olympic games had 16 countries qualify to compete in soccer. In how many different ways can teams of 2 be selected to play each other?

nCr = =

=

120 different ways!

16!2! (16 – 2)!

Permutations (Formula)

(Order does matter! AB is different from BA)

nPr =

Where:n = number of things you can choose

fromr = number you are choosing

n! (n – r)!

Permutations

In the NBA, 8 teams from each conference make the playoffs. They are ranked by their records. If there are 15 teams in the Eastern Conference, how many different ways can they be ordered?

15P8 = =

= 259,459,200 ways to order the teams!

15! (15 – 8)!

15•14•13•12•11•10•9•8•7•6•5•4•3•2•1 7•6•5•4•3•2•1

Combination or Permutation?

1. You choose 3 toppings from 8 possible choices for your pizza.

Combination, 56 different pizzas2. Judges choose the first and second

place winners from 12 projects.Permutation, 132 ways to choose

3. In a class of 22, 3 students will be chosen to go on a field trip.

Combination, 1540 ways to choose

Fundamental Counting Principle

Fundamental Counting Principle

Independent Events – Two or more events whose outcomes have no effect on each other.

The counting principle is used to determine the number of possible outcomes for sequence of independent events.

We are not finding probability, we are simply finding how many different outcomes are possible!

Fundamental Counting Principle

If we have two events (E1 & E2), where E1 can happen n1 different ways and E2 can happen n2 different ways.

The total number of outcomes for these events to occur is n1 • n2.

If we have more than two events, we continue to multiply the number of outcomes to find the total possible outcomes: n1 • n2 • n3 …

SO WE MULTIPLY THE OUTCOMES…

Fundamental Counting Principle

A coin is tossed and a six-sided die is rolled. Find the number of outcomes for the sequence of events.

A coin has 2 outcomes: Heads or TailsA die has 6 outcomes: 1, 2, 3, 4, 5, or 6

2 • 6 = 12 outcomes

Fundamental Counting Principle

For vacation, Ashley packed 5 shirts, 4 pants, and 2 pairs of shoes. How many possible outfits can Angela Create?

5 • 4 • 2 = 40 outcomes

At Fake University there are 5 science, 6 social studies, 4 math, and 7 English classes to choose from. How many possible schedules could you create?

5 • 6 • 4 • 7 = 840 schedules

Coin Example

How many possible outcomes are there if 6 coins are tossed?

2 • 2 • 2 • 2 • 2 • 2 = 64 outcomes

How many possible outcomes are there if 10 coins are tossed?

2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 = 1024 outcomes