Ask Sid about Permutations and Combinations

Clever Sid with a calculator

What are Permutations & Combinations?

Permutations (Order Matters)

Definition: A permutation is an arrangement of objects in a specific order. The order matters in permutations.

For example, The "Combination Lock."

  • It’s actually named wrong—it should be called a Permutation Lock. If your code is 4-2-1, you can’t enter 1-2-4 and expect it to open. The order is everything.
Combination Lock

Combinations (Order Doesn't Matter)

Definition: A combination is a selection of objects where the order does NOT matter. Only the selection itself is important.

For example, choosing 2 letters from A, B, and C:

  • AB, AC, and BC are the only combinations
  • There are 3 total combinations - note that AB is the same as BA
  • Formula: C(3,2) = 3!/(2!×1!) = 6/(2×1) = 3
Choosing pizza toppings

Try These Real-World Examples

Click any example below to automatically calculate it:

Shuffling a Deck of Cards

How many ways can you arrange all 52 cards in a deck?

52! arrangements

≈ 8.07 × 1067 possibilities

Fun fact:

  • If every person on Earth (8 billion people) shuffled a deck every second since the universe began (13.8 billion years ago), we still wouldn't have seen all possible arrangements

UK National Lottery

Choose 6 numbers from 59 - order doesn't matter:

C(59, 6)

45,057,474 combinations

Odds: About 1 in 45 million chance of winning the jackpot!

Horse Racing Exacta

10 horses - predict which finishes 1st and 2nd in exact order:

P(10, 2)

90 possible outcomes

Betting: An exacta pays more than a win bet precisely because order matters!

6-Match Football Accumulator

Each match has 3 outcomes (Home/Draw/Away)

36

729 possible outcome sequences

Why the high payout? Just like the number of possibilities multiplies with every match, the odds multiply too. You are beating a 1-in-729 event!

Premier League Top 4

20 teams - predict who finishes 1st, 2nd, 3rd and 4th in exact order:

P(20, 4)

116,280 possible outcomes

Betting: Order matters - finishing 1st vs 4th means Champions League group stage vs qualifiers!

Horse Racing Trifecta

20 horses - predict 1st, 2nd, 3rd in exact order:

P(20, 3)

6,840 possible outcomes

Betting: Order matters - that's why trifecta bets pay well!

Quick Guide:

Use Permutations when: Order matters (races, passwords, seating arrangements)

Use Combinations when: Order doesn't matter (lottery, team selection, choosing items)

Live Calculator

Input Values
Please enter a value between 0 and 170
Maximum value: 170 (due to JavaScript number limits)
r must be less than or equal to n
Results

Enter values and click Calculate to see results

Growth Visualization
Comparison Chart

The Formulas

1. Permutations without Repetition: P(n,r)

When you select r objects from n total objects, and the order matters:

P(n, r) = n! / (n - r)!

Where n! (n factorial) = n × (n-1) × (n-2) × ... × 2 × 1

Example: Choosing 3 people from 5 for 1st, 2nd, 3rd place: P(5,3) = 5!/(5-3)! = 120/2 = 60 ways

2. Combinations: C(n,r)

When you select r objects from n total objects, and the order does NOT matter:

C(n, r) = n! / (r! × (n - r)!)

Also written as "n choose r" or ⁿCᵣ

Example: UK Lottery - choosing 6 numbers from 59: C(59,6) = 59!/(6!×53!) = 45,057,474 combinations

3. Permutations with Repetition

When you can select the same item multiple times:

nr

Where n = number of options, r = number of selections

Example: A 4-digit PIN with digits 0-9: 104 = 10,000 possibilities

4. Factorial: n!

The number of ways to arrange all n objects:

n! = n × (n-1) × (n-2) × ... × 2 × 1

Note: 0! = 1 by definition

Important Notes

Key Differences:
  • Permutations vs Combinations: Permutations care about order (ABC ≠ BAC), combinations don't (ABC = BAC = CBA)
  • With vs Without Repetition: With repetition allows reusing items (only for permutations), without repetition doesn't
  • Relationship: For the same n and r, permutations ≥ combinations. P(n,r) = C(n,r) × r!
  • Factorial Growth: Factorials grow extremely fast! 10! = 3,628,800 and 20! = 2.4 × 1018
When to Use Which?

Use Permutations when:

  • The order of selection matters
  • Examples: Race positions, passwords, seating arrangements, phone numbers

Use Combinations when:

  • The order of selection does NOT matter
  • Examples: Lottery tickets, team selection, choosing toppings, committee formation
Limitations:
  • Maximum n value is 170 as computers don't want to cope with more than that. The numbers just get silly.
  • Results larger than 10308 will display as "Infinity"
  • For very large calculations, consider using specialized mathematical software
  • Combinations with repetition are not included in this calculator (less common in practice)