To calculate the probability of a combination, you will need to consider the number of favorable outcomes over the number of total outcomes. Combinations are used to calculate events where order does not matter. In this lesson, we will explore the connection between these two essential topics.

## Combinations

*Note: The formulas in this lesson assume that we have no replacement, which means items cannot be repeated.***Combinations** are a way to calculate the total outcomes of an event where order of the outcomes does not matter. To calculate combinations, we will use the formula *n*C*r* = *n*! / *r*! * (*n* – *r*)!, where *n* represents the total number of items, and *r* represents the number of items being chosen at a time.

To calculate a combination, you will need to calculate a factorial. A **factorial** is the product of all the positive integers equal to and less than your number. A factorial is written as the number followed by an exclamation point. For example, to write the factorial of 4, you would write 4!. To calculate the factorial of 4, you would multiply all of the positive integers equal to and less than 4.

So, 4! = 4 * 3 * 2 * 1. By multiplying these numbers together, we can find that 4! = 24.Let’s look at another example of how we would write and solve the factorial of 9.

The factorial of 9 would be written as 9!. To calculate 9!, we would multiply 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1, and that equals 362,880.

## Combinations Formula

Looking at the equation to calculate combinations, you can see that factorials are used throughout the formula. Remember, the formula to calculate combinations is *n*C*r* = *n*! / *r*! * (*n* – *r*)!, where *n* represents the number of items, and *r* represents the number of items being chosen at a time. Let’s look at an example of how to calculate a combination.

There are ten new movies out to rent this week on DVD. John wants to select three movies to watch this weekend. How many combinations of movies can he select?In this problem, John is choosing three movies from the ten new releases. 10 would represent the *n* variable, and 3 would represent the *r* variable. So, our equation would look like 10C3 = 10! / 3! * (10 – 3)!.The first step that needs to be done is to subtract 10 minus 3 on the bottom of this equation. 10 – 3 = 7, so our equation looks like 10! / 3! * 7!.

Next, we need to expand each of our factorials. 10! would equal 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 on the top, and 3! * 7! would be 3 * 2 * 1 * 7 * 6 * 5 * 4 * 3 * 2 * 1. The easiest way to work this problem is to cancel out like terms. We can see that there is a 7, 6, 5, 4, 3, 2 and 1 on both the top and bottom of our equation. These terms can be cancelled out. We now see that our equation has 10 * 9 * 8 left on top and 3 * 2 * 1 left on bottom. From here, we can just multiply.

10 * 9 * 8 = 720, and 3 * 2 * 1 = 6. So, our equation is now 720 / 6.To finish this problem, we will divide 720 by 6, and we get 120. John now knows that he could select 120 different combinations of new-release movies this week.

## Probability

To calculate the probability of an event occurring, we will use the formula: number of favorable outcomes / the number of total outcomes.Let’s look at an example of how to calculate the probability of an event occurring.

At the checkout in the DVD store, John also purchased a bag of gumballs. In the bag of gumballs, there were five red, three green, four white and eight yellow gumballs. What is the probability that John drawing at random will select a yellow gumball?John knows that if he adds all the gumballs together, there are 20 gumballs in the bag. So, the number of total outcomes is 20.

John also knows that there are eight yellow gumballs, which would represent the number of favorable outcomes. So, the probability of selecting a yellow gumball at random from the bag is 8 out of 20.All fractions, however, must be simplified. So, both 8 and 20 will divide by 4. So, 8/20 would reduce to 2/5. John knows that probability of him selecting a yellow gumball from the bag at random is 2/5.

## Probability of Combinations

To calculate the number of total outcomes and favorable outcomes, you might have to calculate a combination. Remember, a combination is a way to calculate events where order does not matter.Let’s look at an example. To enjoy his movies, John decides to order a pizza. Looking at the menu, John sees the Pizza King offers eight different topping (four meat and four vegetables).

The toppings are: pepperoni, ham, bacon, sausage, peppers, mushrooms, onions and olives. John has a coupon for a 3-topping pizza. Choosing ingredients at random, what is the probability of John selecting a pizza with meat only?John is looking for the probability of selecting a meat-only pizza. In order to do so, he will need to calculate the total number of favorable outcomes over the total outcomes possible. Let’s first calculate the total number of outcomes. To calculate the total outcomes, we will use the formula for combinations because the order of the pizza toppings does not matter.

The formula for combinations is *n*C*r* = *n*! / *r*! * (*n* – *r*)!, where *n* represents the number of items, and *r* represents the number of items being chosen at a time.John is selecting three toppings from the eight offered by Pizza King. 8 would represent our *n* term, and 3 would represent our *r* term.

So, our equation will look like 8C3 = 8! / 3! * (8 – 3)!.To solve this equation, we need to subtract 8 – 3 = 5. So, our equation now looks like 8! / 3! * 5!. Next, we need to expand each of these factorials. 8! would equal 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1, divided by 3! * 5!, which would equal 3 * 2 * 1 * 5 * 4 * 3 * 2 * 1.

Remember, to make this problem easier, we can cancel out like terms on both the top and bottom of this equation. We can see that there is a 5, 4, 3, 2 and 1 on both the top and bottom. These terms can be cancelled out.

By multiplying 8 * 7 * 6 on top, it equals 336. On the bottom, 3 * 2 *1, which equals 6. After dividing 336 by 6, we can see the total number of outcomes of pizzas that John can order is 56.John must now find the number of favorable outcomes. John wanted to know the probability of selecting a pizza with meat only. Looking at the menu, we can see that there are four types of meat to choose from, and John is only selecting three.

This is another example of a combination problem because the order that the meat toppings are selected does not matter. To calculate the number of favorable outcomes, we need to use the combination formula *n*C*r* = *n*! / *r*! * (*n* – *r*)!, where *n* represents the number of items, and *r* represents the number of items being chosen at a time.Since there are four meats and John is choosing three, the *n* term would be 4, and the *r* term would be 3.

Our equation would look like 4C3 = 4! / 3! * (4 – 3)!. Next, we need to subtract the 4 – 3 on bottom, which equals 1. So, our equation now looks like 4! / 3! * 1!.

Next, let’s expand both the top and the bottom of our equation now to look for common terms that we can cancel out. By expanding the top, we get 4 * 3 * 2 * 1, and the bottom would be 3 * 2 * 1 * 1. We can see that on both the top and bottom, there is a 3 * 2 * 1, which can be cancelled out.John can now see that there are only four combinations of 3-topping pizzas that would contain only meats. To calculate the probability, John will need to use the number of favorable outcomes, which was 4, over the number of total outcomes, which was 56. The probability would be 4/56, which can be reduced to 1/14. So, the probability that John selects a 3-topping pizza that will contain only meat is 1/14.

## Lesson Summary

Remember that **combinations** are a way to calculate the total outcomes of an event where order of the outcomes does not matter. To calculate combinations, we will use the formula *n*C*r* = *n*! / *r*! * (*n* – *r*)!, where *n* represents the number of items, and *r* represents the number of items being chosen at a time.To find the probability of an event, you may have to find the combinations. To calculate the probability of an event occurring, you will use the formula number of favorable outcomes over the number of total outcomes.

## Learning Outcomes

After you’ve finished with this lesson, you’ll be able to:

- Identify the formulas to calculate combinations and probabilities
- Calculate factorials
- Solve probability problems that also contain combinations