Fundamental Counting Principal Good Morning happy Tuesday!

Counting Outcomes and Tree Diagrams Tree diagrams are used to help visualize and count outcomes. Example Tossing a coin 3 times

Tree Diagrams expanded Tree diagrams can be used to organize information and count outcomes Any path from left to right through the tree diagram is an outcome

Tree Diagrams Applied when actions are sequential Root of the tree corresponds to the starting point Branches connect the root to nodes representing different outcomes Process continues until all stages are complete

Example A fast food restaurant sells hot dogs, hamburgers, chicken sandwiches, and barbecue sandwiches. They offer as sides french fries, hushpuppies, or onion rings. How many possible combinations are there? Create a tree diagram to show this.

You try! A mechanic offers three types oil changes (standard, synthetic, and high mileage), two types of wiper blades (low profile, and heavy use) and two types of mufflers (chrome and matte black). What are the possible combinations? Make a Tree Diagram to show possible outcomes.

Fundamental Counting Principle Suppose that two events occur in order The first can occur in m ways The second can occur in n ways Then the two events can occur in m x n ways

Example An ice cream store offers three types of cones and 31 flavors. How many different single- cone ice-cream cones is it possible to buy at this store? Types of cones? 3 Flavors of ice cream? 31

Solution 3 types of cones 31 flavors of ice cream By the Fundamental Counting Principle, there are 3 x 31 = 93 ways of choosing a single-scoop ice- cream cone at this store

Example In a certain state, automobile license plates display three letters followed by three digits. How many such plates are possible if repetition of the letter is allowed? How many choices for each letter? 26 How many choices for each digit? 10

Solution 26 10 By the Fundamental Counting Principle, the number of possible license plates is… 26 x 26 x 26 x 10 x 10 x 10 17, 576, 000

How many choices for the first letter? Example In a certain state, automobile license plates display three letters followed by three digits. How many such plates are possible if repetition of the letters is not allowed? How many choices for the first letter? 26 How many choices for the second letter? 25 How many choices for the third letter? 24

Solution 26 25 24 10 By the Fundamental Counting Principle, the number of possible license plates is… 26 x 25 x 24 x 10 x 10 x 10 15, 600, 000

Factorial Notation Product of the first n natural numbers Denoted by n! and is called n factorial n! = n x (n – 1) x (n – 2) x … x 1 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 0! = 1

Example In how many different ways can a race with six runners be completed? Assume there is no tie.

Solution Six possible choices for first place Five possible choices for second place Four choices for third place and so on… So, by the Fundamental Counting Principle the number of different ways the race can be completed is… 6 x 5 x 4 x 3 x 2 x 1 = 6! = 720