Math 141 Lecture Notes
Section 6.3 The Multiplication Principle
The Fundamental Principle of Counting
The Multiplication Principle:
Suppose there are m ways of performing a task T1 and n ways of performing a task T2. Then, there are mn ways of performing the task T1 followed by the task T2.
Example 1: Three trunk roads connect town A and town B, and two trunk roads connect town B and town C.
a. Use the multiplication principle to find the number of ways a journey from town A to town C via town B may be completed.
b. Verify part a. directly by exhibiting all possible routes.
Example 2: Diners at Angelo’s Spaghetti Bar may select their entrees from 6 varieties of pasta and 28 choices of sauce. How many such combinations are there that consist of 1 variety of pasta and 1 kind of sauce?
Example 3: A coin is tossed 3 times, and the sequence of heads and tails is recorded.
a. Determine the number of outcomes of this activity.
b. Exhibit all the sequences by means of a tree diagram.
Example 4: A combination lock is unlocked by dialing a sequence of numbers, first to the left, then to the right, and to the left again. If there are ten digits on the dial, determine the number of possible combinations.
Example 5: An investor has decided to purchase shares in the stock of three companies: one engaged in aerospace activities, one involved in energy development, and one involved in electronics. After some research, the account executive of a brokerage firm has recommended that the investor consider stock from five aerospace companies, three energy development companies, and four electronics companies. In how many ways may the investor select the group of three companies from the executive’s list?
Example 6: Tom is planning to leave for New York City from Washington, D.C., on Monday morning and has decided that he will either fly or take the train. There are five flights and two trains departing for New York City from Washington that morning. When he returns on Sunday afternoon, Tom plans to either fly or hitch a ride with a friend. There are two flights departing from New York City to Washington that afternoon. In how many ways can Tom complete this round trip?