# Sequences, Probability, and Counting Theory

## Counting Principles


### Learning Objectives

1. Solve counting problems using the addition principle.
2. Solve counting problems using the multiplication principle.

### Objective 1: Solve counting problems using the addition principle.

In probability theory, an outcome is a possible result of an experiment or trial.

In probability theory, an event is a set of outcomes of an experiment.

Disjoint events cannot happen at the same time. In other words, they are mutually exclusive.

The addition principle applies when we are making only one selection.


### Practice Makes Perfect


Solve counting problems using the addition principle.



### Objective 2: Solve counting problems using the multiplication principle.

The Multiplication Principle applies when we are making more than one selection.


### Practice Makes Perfect


Solve counting problems using the multiplication principle.


A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!
We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.

### Using the Addition Principle

The company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. 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. By the Addition Principle, there are 8 total options, as we can see in .

### Using the Multiplication Principle

The Multiplication Principle applies when we are making more than one selection. Suppose we are choosing an appetizer, an entrée, and a dessert. If there are 2 appetizer options, 3 entrée options, and 2 dessert options on a fixed-price dinner menu, there are a total of 12 possible choices of one each as shown in the tree diagram in .
The possible choices are:
1. soup, chicken, cake
2. soup, chicken, pudding
3. soup, fish, cake
4. soup, fish, pudding
5. soup, steak, cake
6. soup, steak, pudding
7. salad, chicken, cake
8. salad, chicken, pudding
9. salad, fish, cake
10. salad, fish, pudding
11. salad, steak, cake
12. salad, steak, pudding
We can also find the total number of possible dinners by multiplying.
We could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.

### 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.
A family of five is having portraits taken. Use the Multiplication Principle to find the following.

### Finding the Number of Permutations of n Distinct Objects Using a Formula

For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of 
  objects and we want to choose 
  objects from the set in order, we write 
  Another way to write this is 
  a notation commonly seen on computers and calculators. To calculate 
  we begin by finding 
  the number of ways to line up all 
  objects. We then divide by 
  to cancel out the 
  items that we do not wish to line up.
Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. 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. The number of ways this may be done is 
  Using factorials, we get the same result.
There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.
Note that the formula stills works if we are choosing all 
  objects and placing them in order. In that case we would be dividing by 
  or 
  which we said earlier is equal to 1. So the number of permutations of 
  objects taken 
  at a time is 
  or just 
  
A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.

### Find the Number of Combinations Using the Formula

So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with combinations. A selection of 
  objects from a set of 
  objects where the order does not matter can be written as 
  Just as with permutations, 
  can also be written as 
  In this case, the general formula is as follows.
An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are 
  ways to order 3 paintings. There are 
  or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are not selecting 1 painting. There are 4 paintings we could choose not to select, so there are 4 ways to select 3 of the 4 paintings.

### Finding the Number of Subsets of a Set

We have looked only at combination problems in which we chose exactly 
  objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can be ordered. How many different pizzas are possible?
To answer this question, we need to consider pizzas with any number of toppings. There is 
  way to order a pizza with no toppings. There are 
  ways to order a pizza with exactly one topping. If we continue this process, we get
There are 32 possible pizzas. This result is equal to 
  
We are presented with a sequence of choices. For each of the 
  objects we have two choices: include it in the subset or not. So for the whole subset we have made 
  choices, each with two options. So there are a total of 
  possible resulting subsets, all the way from the empty subset, which we obtain when we say “no” each time, to the original set itself, which we obtain when we say “yes” each time.

### Finding the Number of Permutations of n Non-Distinct Objects

We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be  ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the 
  permutations we counted are duplicates. The general formula for this situation is as follows.
In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are 
  ways to order the stars and 
  ways to order the moon.
There are 3,326,400 ways to order the sheet of stickers.

### Key Equations


### Key Concepts

1. If one event can occur in 
  ways and a second event with no common outcomes can occur in 
  ways, then the first or second event can occur in 
  ways. See .
2. If one event can occur in 
  ways and a second event can occur in 
  ways after the first event has occurred, then the two events can occur in 
  ways. See .
3. A permutation is an ordering of 
  objects.
4. If we have a set of 
  objects and we want to choose 
  objects from the set in order, we write 
  
5. Permutation problems can be solved using the Multiplication Principle or the formula for 
  See  and .
6. A selection of objects where the order does not matter is a combination.
7. Given 
  distinct objects, the number of ways to select 
  objects from the set is 
  and can be found using a formula. See .
8. A set containing 
  distinct objects has 
  subsets. See .
9. For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See .

### Section Exercises


### Verbal

For the following exercises, assume that there are 
  ways an event 
  can happen, 
  ways an event 
  can happen, and that 
  are non-overlapping.
Answer the following questions.

### Numeric

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.
For the following exercises, compute the value of the expression.
For the following exercises, find the number of subsets in each given set.
For the following exercises, find the distinct number of arrangements.

### Extensions


### Real-World Applications
