Note that in part c, we found there were 9! ways for 9 people to line up.

There are 362,880 possible permutations for the swimmers to line up. 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. Draw lines for describing each place in the photo.Multiply to find that there are 56 ways for the swimmers to place if Ariel wins first. There are 8 remaining options for second place, and then 7 remaining options for third place. We know Ariel must win first place, so there is only 1 option for first place. Multiply to find that there are 504 ways for the swimmers to place. Once first and second place have been won, there are 7 remaining options for third place. Once someone has won first place, there are 8 remaining options for second place. There are 24 possible permutations of the paintings. There are four options for the first place, so we write a 4 on the first line.Īfter the first place has been filled, there are three options for the second place so we write a 3 on the second line.Īfter the second place has been filled, there are two options for the third place so we write a 2 on the third line. We can draw three lines to represent the three places on the wall. 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. That enables us to determine the number of each option so we can multiply.

To solve permutation problems, it is often helpful to draw line segments for each option. An ordering of objects is called a permutation.įinding the Number of Permutations of n Distinct Objects Using the Multiplication Principle We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. One type of problem involves placing objects in order. The Multiplication Principle can be used to solve a variety of problem types. There are 60 possible breakfast specials. By the Addition Principle, there are 8 total options.

The Addition Principle tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. There are 3 supported tablet models and 5 supported smartphone models. The company that sells customizable cases offers cases for tablets and smartphones. Permutations Using the Addition Principle We will examine this type of mathematics in this section. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. There is a branch of mathematics devoted to the study of counting problems such as this one. We encounter a wide variety of counting problems every day.

Counting the possibilities is challenging! The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. The customer can choose the order of the images and the letters in the monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A new company sells customizable cases for tablets and smartphones.

Math counting principle