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Fundamental Counting Principle

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Fundamental Counting Principle. 5 ! = 5 x 4 x 3 x 2 x 1 = 120. PR-L4 Objectives: To solve probability problems using formulas and calculations rather than sample spaces or tree diagrams. Learning Outcome B-4.

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Fundamental Counting Principle

5 ! = 5 x 4 x 3 x 2 x 1 = 120

PR-L4 Objectives:To solve probability problems using formulas and calculations rather than sample spaces or tree diagrams.

Learning Outcome B-4

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A group of 12 students on a tour are planning the evening's activities. Each student must select one restaurant out of five, and then one entertainment activity out of 4. How many different student-restaurant-entertainment arrangements are possible?We could draw a tree diagram or write a sample space to describe the above situation, but this would be difficult because there are too many outcomes. In this lesson, we will learn how to determine the total number of outcomes without counting them.

Theory – Intro

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A certain brand of bicycle is available in three colours (yellow, blue, and red), two tire types (smooth, grip), and two different seats (economy, custom). How many different bicycles could be bought?We could determine the answer by drawing a tree diagram or writing the sample space and counting the outcomes. The sample space is shown below. The first letter refers to the colour, the second letter to the tire type, and the third letter to the type of seat. The three letters RGC would indicate a Red color - Grip tire - Custom seat.

From the sample space, we see that 12 different bicycles are possible.

Theory – Counting Outcomes

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We could also calculate the total number of models available in the following manner. For each of the three ways to select the colour, there are two ways to select the tires, and for each way to select the tires, there are two ways to select the seat. The total number of ways you could select the bicycle are:# of colours x # of tires x # of seats = # of options

3 x 2 x 2 = 12 waysWe have multiplied the number of ways each choice can be made, and the product is the number of choices.

Theory – Calculating Outcomes

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When two or more choices must be made together, the total number of outcomes can be determined without listing and counting them. The rule for this is known as "The Fundamental Counting Principle."The Fundamental Counting Principle states the following: If one event can occur in 'a' ways, a second event in 'b' ways, a third event in 'c' ways, and so on, then the number of ways that all events can occur one after the other is the product a*b*c. . .

Theory – Fundamental Counting Principle

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Sue has four pairs of shoes, five pairs of jeans, and seven sweaters. How many different clothing combinations can she select?A useful technique to use when solving this question is to draw spaces to represent the number of events, and then writing the number of ways each event can occur. The three events are shoes, jeans, and sweater:and the number of ways each choice can be made is written in the space:The total number of clothing combinations possible is:

Example

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9 x 9 x 9 x 9 = 6561
  • 9 x 8 x 7 x 6 = 3024
  • 8 x 7 x 6 x 4 = 1344

Practice

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How many ways can the letters of the word MONKEY be arranged?Solution:There are six events (and, therefore, six spaces) and we make the choices as indicated.Therefore, there are:

Theory – Permutations

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Note: When we want to multiply all the natural numbers from a particular number down to 1, we can use factorial notation to indicate this operation. The symbol "!" is used to indicate factorial. This notation can save us the trouble of writing a long list of numbers. For example: 6! means 6 x 5 x 4 x 3 x 2 x 1 = 720

4! = 4 x 3 x 2 x 1 = 24

10! = 3 628 800

1! = 1

and we define 0! = 1

Factorial Notation

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How many five-letter "words" can be made from the alphabet if no letters are repeated? A "word" in this case is any five-letter arrangement with all letters different.Solution:We will use the formulato answer this question. The value of n is 26 (there are 26 objects to choose from) and the value of r is 5 (we pick five letters).The formula may be convenient to use, but it is not essential. You can always find the number of arrangements (permutations) by multiplication as shown below.

Example - Permutations

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In how many unique ways can the letters of the word BUTTER be written?

Practice – Permutations with Repetition

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In how many unique ways can the letters of the word CANADA be written?

In how many unique ways can the letters of the word MARMALADE be written?

Practice – Permutations with Repetition

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In the previous examples, the order in which events occurred was important. We now want to look at counting problems where the order does not matter.A committee of three is formed from five students in a class. How many ways can this be done?Solution:The order is not important because the committee of A, B, and C is the same as the committee including A, C, and B. This type of problem, where a set of objects is selected with no regard to the order, is called a combination.

Example - Combinations

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A committee of three is formed from five students in a class. How many ways can this be done?Solution:We begin the calculation the same way as a permutation.Now we must divide by the number of ways in which the events can be arranged among themselves.The factorial method is very convenient if we are dealing with large numbers and using technology (calculators/spreadsheets).

Example – Combinations cont’d

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How many ways can a committee of five be chosen from a group of 20 people?The order in which the committee members are chosen is not important, and so this is a combination problem. The number of committees can be determined as follows:The following formula may be used to determine the number of combinations if you use a scientific calculator. The n refers to the number of objects from which you choose (in this case the number of people) and r refers to the number of objects that are selected at a time (the number of committee members).

Theory – The Combinations Formula

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