In mathematics and statistics, permutations vs combinations are two different ways to take a set of items or options and create subsets. For example, if you have ten people, how many subsets of three can you make? While permutation and combination seem like synonyms in everyday language, they have distinct definitions mathematically.
- Permutations: The order of outcomes matters.
- Combinations: The order does not matter.
Let’s understand this difference between permutation vs combination in greater detail. And then you’ll learn how to calculate the total number of each.
In some scenarios, the order of outcomes matters. For example, if you have a lock where you need to enter four digits, the order matters. If the correct numbers are 8 3 6 2, you can’t enter the same numbers in any other order (e.g., 6 8 2 3) and expect the lock to open! Hence, that’s a permutation.
Conversely, the order of options doesn’t matter for combinations. Imagine you are making a pizza with pepperoni (P), ham (H), and mushroom (M). It doesn’t matter if you make a PHM, HMP, or MPH pizza. It’s still the same pizza! In this case, the order doesn’t matter, so it is a combination.
In some cases, determining whether something is a permutation vs a combination depends on the conditions specified by the problem you’re solving. For example, selecting five people to be in a group where everyone has the same role is a combination because the order you pick them doesn’t matter.
However, if you’re picking five people and their role depends on when you select them, it’s a permutation because the order makes a difference. For example, the first person might be the leader, the second a note-taker, and so on.
Examples of Permutation vs Combination
|Sequence of numbers for a lock.||Numbers for winning the lottery.|
|Selecting individuals for a team by position.||Selecting children to be members of a class.|
|Picking first, second, and third place.||Picking three finalists.|
Counting Formulas for Permutations and Combinations
Frequently, when you’re working with permutations and combinations, you’ll need to calculate the number of possibilities. Now that you understand the differences between the concepts, let’s look at how you count the number of permutations vs combinations. When you have at least two permutations, it’ll have more possibilities than the corresponding number of combinations.
Why is that the case? The pizza example shows that PHM, HMP, and MHP are all the same combination. However, if we’re using those three letters as a password, they’re three distinct permutations.
For permutations, the order matters. Suppose we need to know the number of possibilities for a four-digit lock. There are 10 possible values, and we’re picking 4. Statisticians state this scenario as “10 choose 4” and denote it as 10P4, where P indicates we’re working with permutations. For this scenario, how many permutations are there?
That depends on whether we allow repeated values or not. Can the lock’s sequence use 1 1 2 3? Or can you only use each digit once?
If the sequence can repeat values, then each of the four digits has ten possibilities (0 – 9). Hence, a four-digit lock has 10 * 10 * 10 * 10 = 104 = 10,000 permutations. However, if values can’t repeat, then there are 10 * 9 * 8 * 7 = 5,040. The decreasing values in the second calculation represent the fewer options available for each subsequent digit when you can’t reuse previous values.
Now, suppose we have 10 pizza toppings, and we’re picking 4 of them. We denote this as 10C4.
This arrangement of having 10 options and choosing 4 is the same as the permutation example except now we’re working in a setting where the order doesn’t matter, which helps illustrate the difference between permutation vs combination. How many combinations are there?
To calculate the number of combinations with repetitions, use the following equation:
- n = the number of options.
- r = the size of each combination.
The exclamation mark (!) represents a factorial. In general, n! equals the product of all numbers up to n. For example, 3! = 3 * 2 * 1 = 6. The exception is 0! = 1, which simplifies equations.
In our example, n = 10 and r = 4 because there are 10 toppings available and we’ll choose 4. Let’s input those value into the equation that allows repetition:
There are 715 combinations when you can repeat ingredients. That means we can get double, triple, or even quadruple pepperoni!
If you can’t repeat values, the formula is the following:
Entering the values into the without repetition equation:
There are 210 combinations when you can’t repeat ingredients. We can only get an ingredient once—no double pepperoni!
As expected, the number of possibilities for the 10C4 example are much lower than for the 10P4 example.
Using Permutation vs Combination to Solve Probability Problems
This overview just scratches the surface of using permutations and combinations. For more detail about each one along with worked examples for solving probability problems, please read my separate articles about: