I found that permutations and combinations are two mathematical concepts that often confuse most of my students. Even though they both involve rearranging elements, the two concepts have fundamental differences that every student should understand to use properly.
So, what is the difference between permutation and combination? A permutation is a way of arranging objects or numbers in order. In contrast, a combination is a method to select objects or numbers from a group of objects or collections so that the order of the objects does not matter.
Although permutation and combination are similar, they have some distinct differences that make them useful for different purposes. Read on to find out what sets permutations and combinations apart from each other.
If, like most students, you are wondering if is statistics harder than calculus, I encourage you to read this article.
What is a Permutation?
A permutation is an arrangement of objects or numbers in a particular order. It can also be thought of as an ordered combination. For example, if you have three letters (ABC), then there are six possible permutations of those three letters: ABC, ACB, BAC, BCA, CAB, and CBA.
The number of possible permutations for a given set of objects is calculated by taking the factorial of the number of objects in the set (n!), where n is the number of objects in the set.
The formula for finding permutation:
So for our example above, 3! = 6 because there are six possible permutations for three letters.
I encourage you to watch the video to learn more about permutation.
What is a Combination?
A combination is an unordered selection or grouping where order does not matter. In our example above with three letters (ABC), there would only be one combination which would be ABC since order does not matter when it comes to combinations. In other words, it doesn’t matter whether you list ABC or CBA since both will yield the same result.
The number of possible combinations for a given set of objects can be calculated using nCr where n is the total number of objects and r is the number of objects chosen from that set (nCr).
The formula for finding a combination:
For our example above, 3C3 = 1 because there’s only one combination out of three letters, regardless of how you arrange them.
If you want to learn more about combinations, I suggest you watch the video below.
Difference between Permutation and Combination
The major difference between permutation and combination is that in permutation, the order matters, whereas in combination, the order doesn’t matter. In other words, combinations refer to pairing values within particular criteria or classifications.
The table below shows a snapshot of the key difference between permutation and combination.
| Key Differences | Permutation | Combination |
| Definition | In probability, the term permutation means how many ways we can arrange or order a particular set. | In probability, the term combinations refer to an arrangement or sequence of outcomes where the order does not matter. |
| How important is the order? | Important | Unimportant |
| Examples of Permutation vs. Combination | – Rearranging alphabets, people, numbers, digits, letters, and colors. – Choosing a team captain, a president from a class. – Selecting first, second, and third place prizewinners. | – Choice of menu, clothes, food, subjects, and team – Selecting three winners. – Selecting three team members from a class debate team. |
| Usages | We use permutations when we need an order or arrangement sequence. | We use combinations to find the number of possible sets or groups that we can form. |
How Do You Identify a Permutation And Combination?
Generally, the easiest way to identify a permutation and combination is if the order doesn’t matter, then use a combination; if the order does matter, you will use a permutation.
Knowing how to differentiate between these two concepts can help you make more informed decisions about your data sets when it comes to probability calculations.
Permutations and combinations may sound similar, but they actually serve different functions when it comes to probability calculations. Permutations are arrangements or orders, whereas combinations are groupings or selections where order does not matter.
Understanding both concepts will give you an edge when studying statistics and probabilities related to your data sets and help inform better decision-making going forward.
The table below contains a list of keywords that help us identify a permutation and combination.
| Essential keywords to identify a combination | Essential keywords to identify a permutation |
| – Select – Combination – Choose – Pick | – Arrangements – Unique – Ordered Ways |
What to read next:
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- 6 Reasons Why Statistics Is So Hard (And helpful tips for studying stats!)
Wrapping Up
The major difference between permutations and combinations is all about understanding order. Permutations take order into account, which means the relative position of each element matters.
On the other hand, combinations do not; all that matters is how many different arrangements there can be with a given set of elements, regardless of their placement within those arrangements.
I encourage you to learn when to use permutations and combinations, as it will help you solve problems quickly and accurately.