How to Calculate the Probability of Combinations

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 nCr = 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 nCr = 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?