12.2 The Fundamental Counting Principle

• Understand the fundamental counting principle.

• Use slot diagrams to organize information in counting problems.

• Know how to solve counting problems with special conditions.

The fundamental counting principle solves problems without listing elements or drawing

tree diagrams.

• The Fundamental Counting Principle (FCP) – 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 … ways.

• How many different outfits can be formed with 3 coats, 5 pairs of pants, 7 shirts, and 4 ties?– 3 x 5 x 7 x 4 = 420 outfits.

Slot diagrams help organize information before applying the fundamental counting

principle.• Sometimes special conditions affect the number of

ways we can perform the various tasks. A useful technique for solving problems such as these is to draw a series of blank spaces to keep track of the number of ways to do each task. This is called a slot diagram.

1st task 2nd task 3rd task 4th task

x x xNumber of ways

Number of ways

Number of ways

Number of ways

To open personal locker you must enter 5 digits from the set 0, 1, 2, …, 9. How many different patterns are possible if

a) Any digits can be used in any position and repetition is allowed? 100,000

possibilities

x x x xany digit any digit any digit any digit any digit

b) Any digits can be used in any position, but repetition is not allowed? 30,240

possibilities

x x x x

any digit can’t repeat can’t repeat can’t repeat can’t repeat

10 10 10 10 10

6 7 8 910

In solving counting problems, consider special conditions first.

• Ten students are in a class. Joe and Mary must sit together in the front row. If there are six chairs in the first row of the classroom, how many different ways can the students be assigned to sit in this row?– To begin this problem, first consider the special

condition.

Seat 1 Seat 2 Seat 3 Seat 4 Seat 5 Seat 6

Joe Mary x x x x

x Joe Mary x x x

x x Joe Mary x x

x x x Joe Mary x

x x x x Joe Mary

There are 5 ways to choose the two seats for Joe and Mary and there are 2 ways of deciding how Joe and Mary will sit within these seat – Joe sits either on the right or on the left. There are eight students left; thus, we have 8 for the 1st remaining seats, 7 for the 2nd seat, 6 for the 3rd seat, and 5 for the last seat. Therefore, the number of ways to seat the students in the first row is:

5 x 2 x 8 x 7 x 6 x 5 = 16,800

4251

0011 0010 1010 1101 0001 0100 1011

Classwork/Homework

• Classwork – Page 695 (5 – 21 odd, 37, 39)

• Homework – Page 695 (6 – 20 even, 38 skip #12)