In this lesson, we’ll learn how to identify combinations of objects. We’ll also look at various formulas that allow us to calculate the number of possible combinations in a given scenario.
Combinations and Permutations
In mathematics, combinations and permutations are normally studied at the same time because they are very similar. For instance, both permutations and combinations are collections of objects. But while a combination is a collection of the objects where the order doesn’t matter, a permutation is an arrangement of a group of objects where the order does matter.
For example, suppose you were to choose four from a group of 20 people: Andrea, Alex, Sophie, and Nathan. If we said you chose Alex, Nathan, Sophie and Andrea, we’d still be talking about the same group of people. As the order in which we name them wouldn’t matter, we’d be referring to a combination of four people.
Now, imagine that your employer just assigned you the following 4-digit employee identification code: 4793. If we changed the order of the digits to 9734, we’d end up with a different ID number. As the order of the digits does matter, 4793 is a permutation of four digits.
Before we get into the combination formulas, we first need to talk about n!, not in an excited fashion, which you may have assumed from the exclamation point, but in a mathematical context. In math, n!, pronounced n factorial, represents the product of all of the integers from n down to 1, as shown in the image below:
For example, 4! = 4 * 3 * 2 * 1 = 24. We’ll be using n! a lot when dealing with permutations and combinations.
When it comes to combination formulas, there are two scenarios we want to consider:
- Repetition is allowed
- Repetition is not allowed
To understand what these scenarios mean, let’s revisit the group of people we discussed at the beginning of the lesson. Imagine that Alex was the first of the four we chose to be in our group. Once we’ve chosen Alex, we can’t choose him again since he’s already in the group. So, in this scenario, repetition is not allowed.
Now, pretend you’re at a sushi bar that offers 11 different types of sushi, from which you can choose three. Suppose the first type of sushi you choose is salmon. As the menu allows you to have 3 pieces of sushi, your second and third choices could also be salmon. In this scenario, repetition is allowed.
It is important to understand whether or not repetition is allowed when determining which formula to use when solving combination problems. The formulas we use when dealing with combinations are described in the image below:
Solving Combination Problems
To solve problems involving combinations, we follow these steps:
- Make sure you are dealing with a combination problem, where order does not matter, and not a permutation problem
- Determine if repetition is allowed
- Use the appropriate formula based on what you found in the second step
- Substitute known numbers for the values in the formula, and perform the operations
Let’s see how these steps apply to the group of four people we chose earlier in the lesson. How many different ways can we choose them?
1. We know the order of the four people doesn’t matter, so we’re dealing with a combination, not a permutation.
2. We also know that, in this scenario, repetition is not allowed.
3. Here, we’re working with a combination that doesn’t allow repetition, so we use this formula:
n! / r!(n – r)!
4. As we’re choosing 4 from 20 people, r = 4 and n = 20. So:
20! / 4!(20 – 4)! = 20! / 4!16! = 20*19*18*17 / 4*3*2*1 = 4,845
There are 4,845 ways to choose a group of four people from 20 people.
Now, lets take another look at our plate of sushi, where repetition is allowed. As we can choose 3 from among 11 types of sushi, how many different plates we could order?
1. We know the order in which we choose our three pieces of sushi doesn’t matter, so again, we’re working with a combination.
2. We already saw that, in this scenario, repetition is allowed.
3. Here, we’re working with a combination formula where repetition is allowed, so we use this formula:
(n + r – 1)! / r!(n – 1)!
4. As we’re choosing 3 from 11 pieces of sushi, n = 11 and r = 3. So:
(11 + 3 – 1)! / 3!(11 – 1)! = 13! / 3!10! = 13*12*11 / 3*2*1 = 286
Therefore, we can combine 3 pieces of sushi in 286 different ways when there are 11 pieces to choose from and repetition is allowed.
A combination is a group of objects in which order does not matter, unlike a permutation, which is an arrangement of a group of objects where the order does matter. Another thing to remember for solving combinations is n! In math, n! is pronounced n factorial and represents the product of all of the integers from n down to 1. An example of this would be: 4! = 4 * 3 * 2 * 1 = 24.
When solving combination problems, your first step is to determine if order matters to make sure you’re dealing with a combination. Next, you determine if repetition is, or is not, allowed.
If repetition is not allowed, use this formula:
n! / r!(n – r)!
If repetition is allowed, use this formula:
(n + r – 1)! / r!(n – 1)!