Permutation and combination are two of the most commonly confused topics in GCSE and A-Level Maths. Both involve selecting items from a group, but they answer completely different questions. The difference between permutation and combination comes down to one thing: does the order matter? This guide explains both clearly with real examples, a comparison table, a memory trick, and a quiz so you never mix them up again.
Permutation is an arrangement of items where the order matters. Choosing first, second, and third place in a race is a permutation because 1st, 2nd, 3rd is different from 3rd, 2nd, 1st. Combination is a selection of items where the order does not matter. Choosing three people for a committee is a combination because it does not matter who was picked first. If order matters, use permutation. If order does not matter, use combination.
Difference Between Permutation and Combination: Comparison Table
| Feature | Permutation | Combination |
|---|---|---|
| Does order matter? | Yes | No |
| What it counts | Arrangements | Selections |
| Example question | How many ways can 3 people finish a race? | How many ways can you choose 3 people from a group? |
| Formula | nPr = n! / (n-r)! | nCr = n! / r!(n-r)! |
| Result size | Always larger or equal | Always smaller or equal |
| Real world use | PIN codes, race results, passwords | Lottery numbers, committees, card hands |
| Key question to ask | Does changing the order give a different outcome? | Does changing the order give the same outcome? |
What is Permutation?
A permutation is an arrangement of items in a specific order. When you calculate permutations, you are counting how many different ordered arrangements are possible from a set of items.
The key question with permutations is always: does swapping the order create a different outcome? If yes, you are dealing with a permutation.
Think about a four-digit PIN code. The digits 1, 2, 3, 4 arranged as 1234 is a completely different PIN from 4321 or 2143. The order matters because a different arrangement gives a different PIN. That is a permutation.
The formula for permutations:
nPr = n! / (n – r)!
Where n is the total number of items and r is how many you are choosing.
Example: How many ways can 5 runners finish in the top 3 positions?
n = 5, r = 3
5P3 = 5! / (5-3)! = 120 / 2 = 60 ways
There are 60 different ordered arrangements for the top 3 positions from 5 runners.
What is Combination?
A combination is a selection of items where the order does not matter. When you calculate combinations, you are counting how many different groups or selections are possible, regardless of the order they are arranged in.
The key question with combinations is: does swapping the order create a different outcome? If no, you are dealing with a combination.
Think about choosing 3 people from a group of 10 to form a committee. It does not matter whether you picked Sarah first or last. The committee of Sarah, James, and Priya is the same committee no matter what order you chose them in. That is a combination.
The formula for combinations:
nCr = n! / r!(n – r)!
Where n is the total number of items and r is how many you are choosing.
Example: How many ways can you choose 3 people from a group of 5?
n = 5, r = 3
5C3 = 5! / 3!(5-3)! = 120 / (6 x 2) = 120 / 12 = 10 ways
Notice that 5P3 gave us 60 but 5C3 gives us only 10. Combinations always give a smaller number because they do not count different orderings of the same group as separate outcomes.
Understanding Factorial
Both formulas use the factorial symbol (!). If you have not seen this before, factorial simply means multiply all whole numbers from that number down to 1.
- 5! = 5 x 4 x 3 x 2 x 1 = 120
- 4! = 4 x 3 x 2 x 1 = 24
- 3! = 3 x 2 x 1 = 6
- 2! = 2 x 1 = 2
- 1! = 1
- 0! = 1 (this is a rule, not an intuitive result)
Once you are comfortable with factorials, the permutation and combination formulas become much more manageable.
Example 1 – Race positions (Permutation):
Eight athletes compete in a 100m race. How many different ways can the top 3 finishers be arranged on the podium?
This is a permutation because 1st, 2nd, 3rd matters. Gold is not the same as bronze.
8P3 = 8! / (8-3)! = 8! / 5! = 8 x 7 x 6 = 336 arrangements
Example 2 – Choosing a committee (Combination):
A school needs to choose 4 students from a class of 20 to represent them at a conference. How many different groups are possible?
This is a combination because the group is the same regardless of who was chosen first.
20C4 = 20! / 4!(16)! = 4,845 possible groups
Example 3 – PIN codes (Permutation):
A bank gives customers a 4-digit PIN using the digits 1 to 9 with no repeats. How many different PINs are possible?
This is a permutation because 1234 and 4321 are different PINs.
9P4 = 9! / (9-4)! = 9! / 5! = 9 x 8 x 7 x 6 = 3,024 possible PINs
Example 4 – Lottery (Combination):
A lottery requires you to choose 6 numbers from 1 to 49. How many different combinations of numbers are possible?
This is a combination because the order you pick the numbers does not matter. 1, 7, 14, 23, 38, 42 is the same ticket regardless of the order they were drawn.
49C6 = 13,983,816 possible combinations — which is why winning is so unlikely!
Example 5 – Password creation (Permutation):
A password must use 3 different letters from A, B, C, D, E. How many different passwords are possible?
This is a permutation because ABC and CAB are different passwords.
5P3 = 5! / (5-3)! = 5! / 2! = 60 / 2 = 60 possible passwords
The one question trick:
Every single time you see a counting problem, ask yourself just one question:
“Does the order matter?”
YES = Permutation = Position matters
NO = Combination = just the Collection matters
Another way to remember: think of a combination lock. Despite the name, a combination lock is actually a permutation lock because 1-2-3 and 3-2-1 open different things. It is one of the most famous examples of the word being used incorrectly in everyday life, and it is a great way to remember the difference.
Quick Quiz: Permutation or Combination?
1. You are choosing 2 books to read from a shelf of 8. Does the order matter?
2. A club is awarding 1st, 2nd, and 3rd prizes from 10 members. Does the order matter?
3. A teacher picks 3 students from a class of 25 to work together on a project. Does the order matter?
4. You are creating a 3-letter code from the letters X, Y, Z, W. XYZ and ZYX are considered different codes. Does the order matter?
5. A football manager selects 11 players from a squad of 18. The team is the same regardless of the order players were chosen. Does the order matter?
Difference Between Permutation and Combination in Exams
The difference between permutation and combination is tested regularly in GCSE Maths, A-Level Statistics, and university entrance exams. The most common exam mistake is using the wrong formula because students did not stop to ask whether order matters. Always read the question carefully, identify whether you are arranging or selecting, and then choose the right formula. Showing your working clearly and labelling which formula you are using will also earn you method marks even if your final calculation has a small error.
Common Mistakes to Avoid
Not asking whether order matters:
This is the root cause of almost every permutation and combination mistake. Before writing any formula, stop and ask: does swapping the order change the outcome? If yes, permutation. If no, combination. Make this a habit and you will rarely go wrong.
Forgetting that combinations always give smaller numbers:
nCr is always less than or equal to nPr for the same values of n and r. If your combination answer is larger than your permutation answer for the same problem, something has gone wrong. Use this as a quick sense check.
Mixing up n and r in the formula:
n is always the total number of items available. r is always how many you are choosing. Students sometimes reverse these. Write them out clearly before substituting into the formula.
Forgetting that 0! = 1:
When r equals n in a combination, you end up with 0! in the formula. Remember that 0! = 1 by definition. This catches many students out in exams.
Frequently Asked Questions
What is the relationship between permutation and combination?
Every combination can become multiple permutations by arranging the selected items in different orders. Specifically, nPr = nCr x r!. This means the number of permutations of r items chosen from n equals the number of combinations multiplied by the number of ways to arrange those r items. Combinations count groups. Permutations count ordered arrangements of those groups.
Can permutation and combination give the same answer?
Yes, when r equals 1. Choosing 1 item from a group gives the same result whether you use permutation or combination, because there is only one item to arrange. nP1 = nC1 = n. They also give the same answer when r equals 0, since both equal 1.
Why is a combination lock actually a permutation?
A combination lock requires you to enter numbers in a specific order. 3-7-2 opens the lock but 2-7-3 does not. Since the order matters, it is technically a permutation lock. The name combination lock is mathematically incorrect, but it has stuck in everyday language. It is actually a useful way to remember the difference between the two concepts.
What does nCr mean on a calculator?
nCr is the combination function built into most scientific calculators. You enter the total number of items (n), press the nCr button, then enter how many you are choosing (r), and press equals. Most calculators also have an nPr button for permutations. These buttons do the factorial calculations automatically so you do not have to work them out by hand.
Is permutation or combination used more in real life?
Both appear regularly but in different contexts. Permutations appear in security (passwords, PINs, codes), sports (race results, rankings), and any situation involving ordered arrangements. Combinations appear in probability (card games, lottery), team selection, and any situation where you are forming a group without caring about order. Knowing which applies in a given situation is the most practical skill.
For more Maths help visit Khan Academy: Permutations and Combinations.
Also read: Difference Between Mean, Median and Mode | Difference Between Perimeter and Area | Difference Between Speed and Velocity
The difference between permutation and combination always comes back to that one question: does order matter? Ask it every time and you will always use the right approach. The difference between permutation and combination is one of those concepts that clicks suddenly and then feels obvious. Once it clicks, the formulas make complete sense. The difference between permutation and combination is not about memorising two formulas. It is about understanding what you are actually counting.
The best way to get comfortable with the difference between permutation and combination is to practise with as many real examples as possible. Every time you see a counting problem, stop and ask the order question first. The difference between permutation and combination will start to feel completely natural very quickly once you have that habit in place.
The best way to get comfortable with the difference between permutation and combination is to practise with as many real examples as possible. When you truly understand the difference between permutation and combination, problems that once seemed impossible become straightforward. Every time you see a counting problem, stop and ask the order question first. The difference between permutation and combination will start to feel completely natural very quickly once you have that habit in place.