Counting Techniques. Tree Diagrams Tree Diagrams: A device used to list out all possibilities of a sequence of events in a systematic way. Useful in determining.

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Presentation transcript:

Counting Techniques

Tree Diagrams Tree Diagrams: A device used to list out all possibilities of a sequence of events in a systematic way. Useful in determining the probability of events. Example:

Tree Diagram Examples A sales person has to go from NY to Atlanta to Miami. They can travel from NY to Atlanta by car or bus and then from Atlanta to Miami by train, car, or plane. List all possible ways they can get there. The Rangers and Flyers are playing a best of 3 series. The team to win 2 games wins the series. Create a tree diagram of all possibilities. A quiz has 3 true-false questions. How many possible answer keys are there?

What’s in your closet? Simply write down/type on Chrome Book the amount of favorite pants, favorite shoes, and favorite shirts you have in your closet/dresser.

Multiplication Counting Rules

Multiplication Rules Examples How many id cards can be made if they must have 3 letters followed by 2 numbers in them? How many id cards if we can not repeat letters or numbers? How many ways can 6 people be placed in line? A bag contains 4 different colored chips. 1 chip is chosen at time and then placed back into the bag. How many different color schemes are possible if a chip is taken out 3 times ? How many schemes are there if the chip is not placed back into the bag?

Permutations Permutation: An arrangement of n distinct objects in a specific order. Rule #1: The number of permutations of n distinct objects taken all together is n! Examples: How many ways can 4 books be placed on a shelf? How many ways can 8 people be chosen to stand in a line?

Permutations Rule #2: The arrangement of n objects in a specific order using r objects at a time is written in the format: Examples: How many different ways can a president and vice president be chosen from a group of 6 people? A research company decides to test a product in 5 different areas. How many ways can 3 areas be selected in a certain order for the first test?

Permutations Rule #3:

Examples How many different permutations can be made from the word mathematics? How many different permutations can be made from the word football?

Combinations Combination: A selection of distinct object without regard to the order. Formula: Examples: How many ways can a committee of 3 people be chosen from a group of 9 people? How many ways can 5 cars and 6 trucks be chosen from 8 cars and 10 trucks to be tested for inspection? How many ways can 5 cards be chosen from a deck of 52 cards?