# 13.5 Counting principles  (Page 2/12)

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## The multiplication principle

According to the Multiplication Principle    , if one event can occur in $m$ ways and a second event can occur in $n$ ways after the first event has occurred, then the two events can occur in $m×n$ ways. This is also known as the Fundamental Counting Principle    .

## Using the multiplication principle

Diane packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit and decide whether to wear the sweater. Use the Multiplication Principle to find the total number of possible outfits.

To find the total number of outfits, find the product of the number of skirt options, the number of blouse options, and the number of sweater options.

There are 16 possible outfits.

A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are 3 types of breakfast sandwiches, 4 side dish options, and 5 beverage choices. Find the total number of possible breakfast specials.

There are 60 possible breakfast specials.

## Finding the number of permutations of n Distinct objects

The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation    .

## Finding the number of permutations of n Distinct objects using the multiplication principle

To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.

There are four options for the first place, so we write a 4 on the first line.

After the first place has been filled, there are three options for the second place so we write a 3 on the second line.

After the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally, we find the product.

There are 24 possible permutations of the paintings.

Given $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ distinct options, determine how many permutations there are.

1. Determine how many options there are for the first situation.
2. Determine how many options are left for the second situation.
3. Continue until all of the spots are filled.
4. Multiply the numbers together.

## Finding the number of permutations using the multiplication principle

At a swimming competition, nine swimmers compete in a race.

1. How many ways can they place first, second, and third?
2. How many ways can they place first, second, and third if a swimmer named Ariel wins first place? (Assume there is only one contestant named Ariel.)
3. How many ways can all nine swimmers line up for a photo?
1. Draw lines for each place.

There are 9 options for first place. Once someone has won first place, there are 8 remaining options for second place. Once first and second place have been won, there are 7 remaining options for third place.

Multiply to find that there are 504 ways for the swimmers to place.

2. Draw lines for describing each place.

We know Ariel must win first place, so there is only 1 option for first place. There are 8 remaining options for second place, and then 7 remaining options for third place.

Multiply to find that there are 56 ways for the swimmers to place if Ariel wins first.

3. Draw lines for describing each place in the photo.

There are 9 choices for the first spot, then 8 for the second, 7 for the third, 6 for the fourth, and so on until only 1 person remains for the last spot.

There are 362,880 possible permutations for the swimmers to line up.

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