1.2 Notes/Examples

Please take notes on this material.

Overview of Notes and Examples

Class Notes File

The Fundamental Counting Principle states that if you wish to find the number of outcomes for a given situation, simply multiply the number of outcomes for each individual event. In Example 2 in section 1.1, the child had two different choices of drink and three different choices of meat. If we multiply 2 times 3, we get 6 which is the total number of outcomes possible. The Fundamental Counting Principle expands to any number of events. For example, suppose it turned out that the child also wanted to order eggs and had a choice between scrambled and sunny-side up. The fundamental counting principle states that there are:

2 x 3 x 2

or 12 ways to order this breakfast.

There are other ways to visually see what is happening here. Let's use a tree diagram.

Example 1

Build a tree diagram that shows the different outcomes for what the child might order for breakfast.

Solution

The first set of branches of the tree diagram will represent the type of drink, the second set of branches will represent the type of meat, and the third set of branches will represent the type of egg. A diagram of what this will look like is shown below.

We have labeled the ends of two of the branches in the figure above to show what each branch means. For example, one of the labeled branches shows that the child might have ordered milk, bacon, and scrambled eggs.

The Fundamental Counting Principle is critically important especially when considering complex tree diagrams. Our tree diagram above has many branches and it tracks a great deal of material. It ultimately shows us the 12 different possible breakfast orders, but it takes a large amount of organization to successfully complete. Multiplying 2 by 3 by 2 is a much quicker way to find out the total number of possible outcomes.

Example 2

A couple is planning to have 3 children. Consider the different results that might occur in terms of gender. For example one outcome might be Boy, Boy, Girl (BBG).

a) Using the Fundamental Counting Principle, calculate the number of different outcomes for the children in this family.

b) Build a tree diagram that shows the different orders of children the couple might have.

c) Construct the sample space that shows all the different orders of children the couple might have.

Solution

a) There are 2 choices for the first child, 2 for the second, and 2 for the third. Therefore, there are:

2 X 2 X 2 = 8 outcomes for the gender order of the 3 children.

c) In order to be organized, the list will be alphabetized. BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG There are a total of 8 outcomes.

There are many other ways to apply the Fundamental Counting Principle. A standard deck of cards has 52 cards as shown below. If you are dealt just one card, there are 52 different outcomes.

[Figure4]

Example 3

Suppose you are dealt two cards from a standard deck of 52 cards. How many different outcomes are possible?

Solution

We could certainly try drawing a tree diagram but that could get very large quite quickly. The first split alone would have 52 branches on it. On the other hand, if we use the Fundamental Counting Principle, we can simply calculate how many different ways we could be dealt 2 cards from a standard deck. There would be 52 choices for the 1st card and 51 choices for the 2nd card. (Once the first card is dealt, the deck only has 51 cards left in it.)

There are 52 x 51 = 2652 ways that we could be dealt two cards from the deck.

Example 4

How many different 7-digit phone numbers are possible if no phone number may begin with a zero?

Solution

There are a total of 10 digits available {0, 1, 2, ... 7, 8, 9}. We can't use zero for the first digit so there are only 9 choices for the 1st digit. After that, there are 10 digits available for each of the remaining six digits.

This gives us 9 x 10 x 10 x 10 x 10 x 10 x 10 = 9 x 10^6 = 9,000,000 ways to come up with a 7 digit phone number.

Example 5

A teenager is given 5 different jobs that they must do before they may go out to a movie with friends. The jobs are washing the car, starting a load of laundry, vacuuming the family room, taking out the garbage, and putting away the dishes. In how many different orders could the teenager complete these jobs?

Solution

There are five choices the teenager could pick for the first job. Once that job is finished, there are only 4 jobs remaining. Once the 2nd job is completed, there are only 3 choices for the 3rd job. Once the 3rd job is finished, there are only 2 choices for the 4th job and finally there will only be one choice left for the 5th job.

There are 5 x 4 x 3 x 2 x 1 = 120 different orders that these jobs could be completed. Note that there is a quick way to do this ordered multiplication using factorials. The 5! is read "Five Factorial". Be sure to locate the factorial key on your calculator.

5! = 5 x 4 x 3 x 2 x 1 = 120

8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320

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