Permutations and Combinations- Differences
What Are Permutations and Combinations?
These are two counting techniques that students and professionals mix up constantly. They're not the same thing, and using the wrong one will give you the wrong answer every single time.
Permutations deal with arrangements where order matters. Combinations deal with selections where order doesn't matter. That's it. That's the whole difference.
Everything else flows from this one distinction.
Permutations: When Order Counts
If you're arranging items and switching two things around creates a different result, you're working with permutations.
Example: How many ways can you arrange the letters A, B, C?
ABC, ACB, BAC, BCA, CAB, CBA. That's 6 arrangements.
Swap A and B in ABC to get BAC — that's a completely different arrangement. Order matters here.
Permutation Formula
The formula for permutations is:
P(n,r) = n! / (n-r)!
Where n is the total number of items and r is how many you're arranging.
For P(5,3) = 5! / (5-3)! = 120 / 2 = 60 arrangements
Combinations: When Order Doesn't Count
If you're just picking items and the arrangement doesn't create a new result, you're working with combinations.
Example: How many ways can you choose 3 people from a group of 5 for a committee?
Pick Alice, Bob, Carol. Pick Carol, Alice, Bob. Pick Bob, Carol, Alice. Same committee. These aren't different outcomes — they're the same selection counted once.
That's why combinations always give you a smaller number than permutations for the same inputs.
Combination Formula
The formula for combinations is:
C(n,r) = n! / r!(n-r)!
Notice the extra r! in the denominator compared to permutations. This divides out the arrangements of your selected items.
For C(5,3) = 5! / 3!(5-3)! = 120 / 6(2) = 10 ways
Side-by-Side Comparison
| Feature | Permutations | Combinations |
|---|---|---|
| Order | Matters | Doesn't matter |
| Formula | n! / (n-r)! | n! / r!(n-r)! |
| Result size | Larger number | Smaller number |
| Think of it as | Arrangements | Selections |
| Real-world use | Seating arrangements, passwords, race finishes | Committees, lottery draws, team selections |
How to Decide Which One to Use
Ask yourself one question before you start solving:
"Does swapping two items create a different outcome?"
- If yes → Permutation
- If no → Combination
That's the test. Apply it every time and you won't go wrong.
Practical Examples
Permutation Example: Passwords
How many 4-digit passwords can you create using the digits 1-6 if digits can repeat?
6 × 6 × 6 × 6 = 1,296 passwords
Why? The password 1234 is completely different from 4321. Order absolutely matters here.
Combination Example: Poker Hands
How many 5-card hands can be dealt from a 52-card deck?
C(52,5) = 2,598,960 hands
Why? A hand of Ace-King-Queen-Jack-10 is the same regardless of the order you received the cards. You're just selecting 5 cards.
Permutation Example: Race Results
In a race with 8 runners, how many ways can first, second, and third place be determined?
P(8,3) = 8! / 5! = 336 ways
Why? Finishing 1st-2nd-3rd is completely different from 3rd-2nd-1st. The order of finish matters.
Common Mistakes to Avoid
- Assuming order matters when it doesn't — always check your scenario
- Forgetting the r! in the combination formula — this is the most common error
- Mixing up the formulas — permutation has (n-r)! only, combination has r!(n-r)!
- Overcounting in combinations — remember that ABC, ACB, and BAC are all the same selection
Quick Reference
Use this mental shortcut:
- Permutation = "Arrangements" or "Ordered selections"
- Combination = "Selections" or "Groups"
When in doubt, write out a small example by hand. If you can list all the possibilities without formulas and see duplicates that should count as one — that's a combination. If every swap creates something new — that's a permutation.