Want to cite, share, or modify this book? This book uses the Using factorials, we get the same result. The number of ways this may be done is 6 × 5 × 4 = 120. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. They need to elect a president, a vice president, and a treasurer. Let’s see how this works with a simple example. We then divide by ( n − r ) ! ( n − r ) ! to cancel out the ( n − r ) ( n − r ) items that we do not wish to line up. To calculate P ( n, r ), P ( n, r ), we begin by finding n !, n !, the number of ways to line up all n n objects. Another way to write this is n P r, n P r, a notation commonly seen on computers and calculators. If we have a set of n n objects and we want to choose r r objects from the set in order, we write P ( n, r ). Before we learn the formula, let’s look at two common notations for permutations. Fortunately, we can solve these problems using a formula. How many ways can the family line up for the portrait if the parents are required to stand on each end? Finding the Number of Permutations of n Distinct Objects Using a Formulaįor some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. 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. Finding the Number of Permutations of n Distinct Objects Using the Multiplication Principle An ordering of objects is called a permutation. 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. Finding the Number of Permutations of n Distinct Objects Find the total number of possible breakfast specials. There are 3 types of breakfast sandwiches, 4 side dish options, and 5 beverage choices. By the Addition Principle, there are 8 total options, as we can see in Figure 1.Ī restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. 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. 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. Solve counting problems using permutations involving n non-distinct objects.Ī new company sells customizable cases for tablets and smartphones.Find the number of subsets of a given set.Solve counting problems using combinations.Solve counting problems using permutations involving n distinct objects.Solve counting problems using the Multiplication Principle.Solve counting problems using the Addition Principle.Stated simply, it is the idea that if we have a ways of doing something and b ways of doing another thing and we can not do both at the same time, then there are a + b ways to choose one of the actions. The rule of sum is another basic counting principle.
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