Counting Techniques

0.4

Fundamental Counting Principal

• TWO EVENTS: If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is mn.

• Ex: If one event can occur in 2 ways and another event can occur in 6 ways, then both events can occur in 26 = 12 ways.

Example 1

• At Oswego East High School, there are 603 freshmen, 470 sophomores, 446 juniors, and 292 seniors. In how many different ways can a committee of 1 person from each class be chosen?

Example 2

• Mr. and Mrs. Cal Q. Leight go out to dinner after a long day sitting on their elliptical deck and enjoying a swim in their pool. At the restaurant they go to, they each have a choice of 8 different entrees, 2 different salads, 12 different soft drinks, and 6 different desserts. In how many ways can Mr. C choose 1 salad, 1 entrée, 1 soft drink, and 1 dessert?

Fundamental Counting Principle with Repetition

Example 3: The standard configuration of an Illinois licenseplate use to be 3 letters followed by 3 digits.

a) How many different license plates are possible if letters and digits can be repeated?

b) How many different license plates are possible if letters and digits cannot be repeated?

Example 4

• A multiple choice test has 10 questions with 4 multiple choices for each question. In how many different ways could you complete the test?

Permutations

• Permutation: an ordering of n objects, where the order of the objects does matter.

• EX: There are 6 permutations of the letters A, B, and C: ABC, ACB, BCA, BAC, CAB, and CBA.

• Since there are 3 choices for the 1st letter, 2 choices for the 2nd, and 1 choice for the 3rd, there are 3 2 1 = 6 ways to arrange the letters.∙ ∙

• (3 2 1 can be written 3!) ∙ ∙

In general, the number of permutations of n objects is n!

Example 5

10 skiers are competing in the final round ofthe Olympic freestyle skiing aerial competition.

a) In how many different ways can the skiers finish the competition? (Assume there are no ties)

b) In how many different ways can 3 of the skiers finish first, second, and third to win the gold, silver, and bronze?

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 denoted by

!!

rn

nPrn

Example 6

There are 8 movies you would liketo see that are currently showing intheaters, you movie-goer!• a) In how many different ways can you see all

of the 8 movies?

• b) In how many ways can you choose a movie to see this Saturday and one to see this Sunday?

Combinations

• a selection of r objects from a group on n objects where the order is not important.

)!rn(!r

!nCrn

Example 7:

• a) Find the # of ways to purchase 3 different kinds of juice from a selection of 10 different juices.

• b) Find the # of ways to rent 4 comedy DVDs from a collection of 9 comedy DVDs.

• c) Find the # of ways 3 students can be selected from a committee of 5.

120

126

10

Permutation or combination?

*Permutations: ORDER MATTERS*Combinations: ORDER DOES NOT MATTER

Example 3: Decide if the situation is a permutation or combination.

a) 4 recipes were selected for publication and 302 recipes were submitted.

b) 4 out of 200 contestants were awarded prizes of $100, $75, $50, and $25.

c) A president and vice-president are elected for a class of 210 students.d) Nine players are selected for a team of 15 to start the baseball game.e) The batting order for the 9 starting players is announced.

Warm up

1. There are 25 players on a baseball team. In how many ways can you choose 9 players to start the game?

2. There are 12 books on the summer reading list. You want to read some or all of them. In how many orders can you read…

a) 4 of the books? b) all 12 of the books?

Example 8:

A movie theater has 14 different movies showing. If you want to attend at least 12 of the movies, how many different combinations can you attend?

At least and At most

Example 9:

An ice cream shop has a choice of 10 toppings. Suppose you can afford at most four toppings. How many different types of ice cream sundaes can you order?

At least and At most

Deck of Cards

• How many cards are in a standard deck?• How many suits and what are they?• How many cards in each suit?• How many face cards?

• How many kinds are there?• How many of each kind are there?

Deck of cards: when dealt a hand, the order you receive the cards does not matter.

a) Using a standard deck of 52 cards, how many different 7-card hands are possible?

b) How many of these 7 card hands have 2 jacks, 3 sevens, and 2 aces?

Example 10

c) How many of these 7 card hands have 4 kings and 3 other cards?

Example 10

e) How many possible 5 card hands contain exactly 3 kings?

Example 10