Chapter 13 sec 2

 Form your groups, Please.

What is the FCP?  Definition: b  If we want to perform a series of tasks and the first task can be done in a ways, the second can be done in b ways, the third can be done in c ways, and so on, then all the tasks can be done in a x b x c x d x ……. a x b x c x d x ……. ways.

Coin and dice problems.  How many ways can four coins be flipped?  Solution:  There are 4 tasks, flipping the first coin in 2 ways. Therefore the second, third, & fourth also can be flipped 2 ways. Using FCP…  2 x 2 x 2 x 2 = 16 ways

How many ways can four dice be rolled?  Solution:  The first die can be rolled 6 ways. Using the FCP…  6 x 6 x 6 x 6 = 1296 ways.

Methods of FCP  1. Using the slot diagrams.  2. Using the tree diagrams. This chapter focuses on the slot diagram.

Slot Diagram  The slot diagram is a useful technique for solving problems to keep track of the number of ways to do each tasks.

Slot diagram 1 st task 2 nd 3 rd 4 th 5 th x x x x # of ways

Breaking and Entering (forgetfulness)  Your office has a 5 digit keypad lock. The numbers are from 0, 1, …, 9. How many different keypads patterns are possible if …

 a) any digits can be used in any position and repetition of digits is allowed?  b) the digit 0 cannot be using as the first digit, but otherwise any digit can be used in any position and repetition is allowed.  c) any digits can be used in any position, but repetition is not allowed.

Solution A) 5 tasks and there are 10 digits. = 100, 000 ways. 1 st task 2 nd 3 rd 4 th 5 th x x x x 10

 b) 9 x 10 x 10 x 10 x 10 = 90,000 ways.  c) 10 x 9 x 8 x 7 x 6 = 30,240 ways.

Group problems  1. The board of an Internet start-up company has seven members. If one person is to be in charge of marketing and another in charge of research, in how many ways can these two positions be filled?  Solution: 42

 2. If the Chamber of Commerce has 20 members, in how many ways can they elect a different president, vice president, and treasurer?  Solution: 6,840

 Using the digits 0, 1, 2, …, 9, determine the four digit number of possibilities for each situation.  1. The number must begin and end with an odd digit; digits may not be repeated.  Solution: 1,120

 The number must be between 5,001 and 8,000; digits may not be repeated.  Solution: 1,512

 In a certain state, license plates currently consist of two letters followed by three digits. How many such license plates are possible? If the state department of transportation decides to change the plates to have three letters followed by two digits, how many plates will be possible?  Solution: 676,000; 1,757,600