5.4 The Multiplication Principle

1. Tree Diagram2. Multiplication Principle3. Generalized Multiplication Principle

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Tree DiagramA tree diagram is a graph showing possibilities for an event having choices along the way. Suppose a rat in a maze starts at point A. There are five possible routes to get from point A to point B and 3 possible routes to get from point B to the final destination, point C. Represent the possible choices using a tree diagram.

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Tree Diagram (2)

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Possible paths:1a, 1b, 1c2a, 2b, 2c3a, 3b, 3c4a, 4b, 4c5a, 5b, 5cTotal possible paths = 15

Multiplication PrincipleMultiplication Principle Suppose that a task is composed of two consecutive operations. If operation 1 can be performed in m ways and, for each of these, operation 2 can be performed in n ways, then the complete task can be performed in mn ways.

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Example Multiplication Principle

Use the multiplication principle to determine the number of different paths a rat can take going from point A to point C if there are five possible routes to get from point A to point B and 3 possible routes to get from point B to point C. There are 5 possible routes for task 1 and 3 possible routes for task 2 so there are 53 = 15 possible routes. This agrees with the tree diagram drawn previously.

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Generalized Multiplication Principle

Generalized Multiplication Principle Suppose that a task is composed of t operations performed consecutively. Suppose operation 1 can be performed in m1 ways; for each of these, operation 2 can be performed in m2 ways; for each of these, operation 3 can be performed in m3 ways; and so forth. Then the complete task can be performed in

m1m2 m3…mt ways.

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Example Generalized Principle

A corporation has a board of directors consisting of 10 members. The board must select from among its members a chairperson, vice chairperson, and secretary. In how many ways can this be done?

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Example Generalized Principle (2)

The operations are:1. Select chairperson - there are 10 ways (people) to do this.2. Select a vice chairperson - there are 9 ways (9 people left after chairperson is chosen) to do this.3. Select a secretary - there are 8 ways to do this.Total number of ways to complete task is

1098 = 720 ways.8

Summary Section 5.4 The multiplication principle states that the number of ways a sequence of several independent operations can be performed is the product of the number of ways each individual operation can be performed.

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