Math 141: Business Mathematics I Fall 2015 §6.3 The Multiplication Principle Instructor: Yeong-Chyuan Chung The fundamental principle of counting Some problems require more sophisticated counting techniques than those in the previous section. Such problems may involve counting the number of possible combinations or possible arrangements of certain objects. Consider the following problem: A coin is tossed twice, and the sequence of heads and tails is recorded. How many possible outcomes are there? (What if the coin is tossed 100 times?) The following multiplication principle is a fundamental principle of counting. Suppose a task T1 can be performed in N1 ways, a task T2 can be performed in N2 ways,..., and a task Tm can be performed in Nm ways. Then the number of ways of performing the tasks T1 , T2 , . . . , Tm in succession is given by the product N1 N2 · · · Nm . We can think of this in terms of a tree diagram. For example, if m = 3, N1 = 2, N2 = 3, and N3 = 2, then we get the following diagram: 1 2 §6.3 The Multiplication Principle Example (Exercise 6 in the text). Four commuter trains and three express buses depart from City A to City B in the morning, and three commuter trains and three express buses operate on the return trip in the evening. In how many ways can a commuter from City A to City B complete a daily round trip via bus and/or train? Example (Exercise 17 in the text). The Panini Café has a special lunch menu in which the diner chooses a soup or salad, a main course, and a dessert. There are three choices of soups, three choices of salads, five choices for the main course, and four choices of desserts. How many different three-course meals can diners order for lunch? Example (c.f. Exercise 22/26 in the text). An exam consists of ten multiple choice questions with four options each. Assuming that every question is answered, in how many different ways can a student complete the exam (disregarding correctness of answers)? §6.3 The Multiplication Principle 3 Example (Exercise 28 in the text). A password is to be made from a string of five characters chosen from the lowercase letters of the alphabet and the numbers 0 through 9. (a) How many passwords are possible if there are no restrictions? (b) How many passwords are possible if the characters must alternate between letters and numbers? Example (Exercise 30 in the text). A rolling combination four-digit padlock is unlocked by moving each of four rollers so as to produce the correct sequence. Each roller has ten digits. (a) How many possible combinations are there? (b) How many combinations are possible if no digit is repeated? (c) How many combinations are possible if 0 is not the first digit and no digit is repeated? (d) How many combinations are possible if 0 is not the first digit, no digit is repeated, and the four-digit number formed is even?