Fundamental Counting Principle

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  • How many possible pizzas could you make with 3 types of meats, 2 types of cheeses, and 2 types of sauces?
  • 5 * 4 * 3 * 2 * 1 =
  • Fundamental Counting Principle can be used determine the number of possible outcomes when there are two or more characteristics .
  • Fundamental Counting Principle states that if an event has m possible outcomes and another independent event has n possible outcomes, then there are m* n possible outcomes for the two events together.
  • Fundamental Counting Principle
  • Lets start with a simple example.
  • A student is to roll a die and flip a coin. How many possible outcomes will there be?
  • Fundamental Counting Principle
  • For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from?
  • Example 1A: Using the Fundamental Counting Principle
  • To make a yogurt parfait, you choose one flavor of yogurt, one fruit topping, and one nut topping. How many parfait choices are there?
  • Yogurt Parfait
  • (choose 1 of each)
  • Flavor
  • Plain
  • Vanilla
  • Fruit
  • Peaches
  • Strawberries
  • Bananas
  • Raspberries
  • Blueberries
  • Nuts
  • Almonds
  • Peanuts
  • Walnuts
  • Example 1B: Using the Fundamental Counting Principle
  • A password for a site consists of 4 digits followed by 2 letters. Each digit or letter ma be used more than once. How many unique passwords are possible?
  • A permutation is a selection of a group of objects in which order is important.
  • There is one way to
  • arrange one item A.
  • A second item B can
  • be placed first or
  • second.
  • Permutations
  • A Permutation is an arrangement of items in a particular order.
  • Notice, ORDER MATTERS!
  • To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation.
  • You can see that the number of permutations of 3 items is 3 · 2 · 1. You can extend this to permutations of n items, which is n · (n – 1) · (n – 2) · (n – 3) · ... · 1. This expression is called n factorial, and is written as n!.
  • A FACTORIAL is a counting method that uses consecutive whole numbers as factors.
  • The factorial symbol is !
  • Examples 5! = 5x4x3x2x1
  • =
  • 7! = 7x6x5x4x3x2x1
  • =

Factorial !

  • Sometimes you may not want to order an entire set of items. Suppose that you want to select and order 3 people from a group of 7. One way to find possible
  • permutations is to use the Fundamental Counting Principle.
  • First Person
  • Second Person
  • Third Person
  • 7 choices
  • 6 choices
  • 5 choices
  • =
  • arrangements of 4 4!
  • Another way to find the possible permutations is to use factorials. You can divide the total number of arrangements by the number of arrangements that are not used. In the previous slide, there are 7 total people and 4 whose arrangements do not matter.
  • arrangements of 7 = 7! =
  • How many ways can a student government select a president, vice president, secretary, and treasurer from a group of 6 people?
  • Example 2B: Finding Permutations
  • How many ways can a stylist arrange 5 of 8 vases from left to right in a store display?


  • Check It Out! Example 2a
  • Awards are given out at a costume party. How many ways can “most creative,” “silliest,” and “best” costume be awarded to 8 contestants if no one gets more than one award?
  • Permutations
  • A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated?
  • Practice:
  • Combinations
  • A Combination is an arrangement of items in which order does not matter.
  • Since the order does not matter in combinations, there are fewer combinations than permutations.
  • 6 permutations  {ABC, ACB, BAC, BCA, CAB, CBA}
  • 1 combination  {ABC}
  • ***When deciding whether to use permutations or combinations, first decide whether order is important. Use a permutation if order matters and a combination if order does not matter.***
  • Combinations
  • To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible?
  • Practice:
  • Combinations
  • A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions?
  • Practice:


  • There are 12 different-colored cubes in a bag. How many ways can Randall draw a set of 4 cubes from the bag?
  • Check It Out! Example 3
  • The swim team has 8 swimmers. Two swimmers will be selected to swim in the first heat. How many ways can the swimmers be selected?
  • 1. Six different books will be displayed in the library window. How many different arrangements are there?
  • 2. The code for a lock consists of 5 digits. The last number cannot be 0 or 1. How many different codes are possible?
  • 3. The three best essays in a contest will receive gold, silver, and bronze stars. There are 10 essays. In how many ways can the prizes be awarded?
  • 4. In a talent show, the top 3 performers of 15 will advance to the next round. In how many ways can this be done?

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