Math 8 min read

Combinations vs Permutations: When Order Matters (with Examples)

Learn the difference between combinations and permutations, when to use each formula, and how to solve counting problems step by step.

The Quick Answer

Combinations count selections where order does not matter. Permutations count arrangements where order does matter.

Question Type Formula Example
How many 5-card poker hands? Combination C(52, 5) 2,598,960
How many 4-digit PINs (no repeats)? Permutation P(10, 4) 5,040
Choose 3 toppings from 8? Combination C(8, 3) 56
Rank top 3 from 10 contestants? Permutation P(10, 3) 720

The deciding question: does rearranging the same items create a different outcome?

  • Yes → permutation (order matters)
  • No → combination (order does not matter)

The Formulas

Combination: C(n, r)

Choose r items from n total, order irrelevant.

C(n, r) = n! / (r! × (n − r)!)

Permutation: P(n, r)

Arrange r items from n total, order matters.

P(n, r) = n! / (n − r)!

The Relationship Between Them

Notice that C(n, r) = P(n, r) / r!. A combination is a permutation divided by the number of ways to rearrange the chosen items. This makes sense: if you have a group of 3 items, there are 3! = 6 ways to order them. So there are always fewer combinations than permutations for the same n and r.

P(10, 3) = 720
C(10, 3) = 720 / 3! = 720 / 6 = 120

Permutations count every ordering separately. Combinations collapse them into one.

Step-by-Step Examples

Example 1: Lottery Numbers

A lottery draws 6 numbers from 49. How many possible tickets exist?

This is a combination — the numbers {3, 17, 22, 35, 41, 48} win regardless of the order they are drawn.

C(49, 6) = 49! / (6! × 43!)

Simplify by cancelling 43!:
= (49 × 48 × 47 × 46 × 45 × 44) / (6 × 5 × 4 × 3 × 2 × 1)
= 10,068,347,520 / 720
= 13,983,816

There are 13,983,816 possible tickets. Your chance of winning with one ticket is about 1 in 14 million.

Example 2: Race Podium

10 runners compete. How many ways can gold, silver, and bronze be awarded?

This is a permutation — finishing 1st is different from finishing 3rd.

P(10, 3) = 10! / 7!
= 10 × 9 × 8
= 720

There are 720 possible podium results.

Example 3: Pizza Toppings

A menu has 12 toppings. You pick 4. How many different pizzas can you make?

Order does not matter — pepperoni-mushroom-olive-onion is the same pizza as onion-olive-mushroom-pepperoni.

C(12, 4) = 12! / (4! × 8!)
= (12 × 11 × 10 × 9) / (4 × 3 × 2 × 1)
= 11,880 / 24
= 495

There are 495 possible 4-topping pizzas.

Example 4: Passwords from Letters

How many 3-letter codes can you make from 26 letters, if no letter repeats and order matters?

P(26, 3) = 26 × 25 × 24 = 15,600

If order did not matter (just choosing 3 letters):

C(26, 3) = 15,600 / 6 = 2,600

The distinction: "ABC" and "CBA" are different passwords but the same letter selection.

Factorial: The Building Block

Both formulas use factorials. A factorial (written n!) is the product of all positive integers from 1 to n.

5! = 5 × 4 × 3 × 2 × 1 = 120
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320

Special case: 0! = 1 by definition. This is not arbitrary — it makes the formulas work correctly when r = 0 or r = n.

Factorials Grow Extremely Fast

 5! =            120
10! =      3,628,800
15! =  1,307,674,368,000
20! ≈  2.43 × 10¹⁸

By 70!, the number exceeds 10¹⁰⁰ (a googol). This is why lottery odds are so extreme and why brute-force cracking of long passwords is impractical.

How to Tell Combinations from Permutations

Ask these questions:

  1. Am I choosing or arranging? Choosing = combination. Arranging into positions = permutation.
  2. If I swap two items, does the result change? If yes, permutation. If no, combination.
  3. Are the "slots" distinguishable? Named positions (1st place, 2nd place) = permutation. Identical slots (just "chosen") = combination.

Quick Reference

Scenario Type Why
Selecting committee members Combination No ranked positions
Arranging books on a shelf Permutation Position matters (left to right)
Choosing lottery numbers Combination Order of draw is irrelevant
Assigning tasks to people Permutation Each person gets a specific task
Forming a hand of cards Combination Same cards regardless of deal order
Creating a playlist order Permutation Song sequence changes the experience
Picking ingredients for a recipe Combination Same dish regardless of selection order
Seating people around a table Permutation Each seat is different

Common Variations

Combinations with Repetition

What if you can pick the same item more than once? For example, choosing 3 scoops from 5 ice cream flavors, with repeats allowed.

The formula uses a different approach:

C_rep(n, r) = C(n + r − 1, r) = (n + r − 1)! / (r! × (n − 1)!)

For 3 scoops from 5 flavors:

C_rep(5, 3) = C(7, 3) = 35

Permutations with Repetition

If repetition is allowed (like a 4-digit PIN where digits can repeat):

n^r = 10⁴ = 10,000 possible PINs

This is simpler — each slot has n choices independently.

Comparison Table

Type Repetition? Order Matters? Formula
Permutation No Yes n! / (n−r)!
Permutation Yes Yes nʳ
Combination No No n! / (r! × (n−r)!)
Combination Yes No (n+r−1)! / (r! × (n−1)!)

Real-World Applications

Probability

Combinations are the backbone of probability calculations. The probability of an event is:

P(event) = favorable outcomes / total outcomes

Both parts often require counting combinations.

Example: What is the probability of being dealt exactly 2 aces in a 5-card poker hand?

Favorable = C(4, 2) × C(48, 3) = 6 × 17,296 = 103,776
Total     = C(52, 5) = 2,598,960
Probability = 103,776 / 2,598,960 ≈ 3.99%

Sports Brackets

A tournament with 16 teams playing single-elimination has C(16, 2) = 120 possible matchups per round, and the total number of possible bracket outcomes is 2¹⁵ = 32,768.

Security and Passwords

Understanding permutations helps evaluate password strength. A password using 8 characters from 62 possible (a-z, A-Z, 0-9) with repetition allowed:

62⁸ = 218,340,105,584,896 ≈ 2.18 × 10¹⁴

That is roughly 218 trillion possible passwords.

Common Mistakes

  1. Confusing "choose" with "arrange." "Choose 3 from 10" is a combination. "Line up 3 from 10" is a permutation. The verb matters.

  2. Forgetting that C(n, r) = C(n, n−r). Choosing 3 items from 10 is the same count as choosing 7 items from 10, because every group of 3 chosen implies a group of 7 not chosen. C(10, 3) = C(10, 7) = 120.

  3. Using the wrong formula for "with repetition" problems. Standard C(n, r) assumes each item is chosen at most once. If repeats are allowed, you need the stars-and-bars formula.

  4. Overcomplicating the calculation. You rarely need to compute full factorials. Cancel common terms first:

    C(100, 3) = (100 × 99 × 98) / (3 × 2 × 1) = 161,700

    You do not need to compute 100! (a number with 158 digits).

  5. Assuming order always matters or never matters. Read the problem carefully. "How many groups" = combination. "How many arrangements" = permutation.

Try It Yourself

Combination & Permutation Calculator

Enter any values of n and r to instantly calculate C(n, r) and P(n, r) with a full step-by-step factorial breakdown.

Open Calculator

Related Tools

Related Tools

Combination Calculator

Calculate nCr combinations and nPr permutations

Probability Calculator

Calculate probability for AND, OR, NOT, and conditional events

Percentage Calculator

Calculate percentages, find X% of Y, percentage change