Permutation and Combination Calculator
Free Permutation & Combination Calculator — nPr & nCr Online (2026)
How many 4-digit PIN codes are possible? How many ways can a cricket team be selected from 15 players? These questions require either permutations or combinations, and the distinction between them is critical.
KEY TAKEAWAYS
- Permutation (nPr): order matters. Combination (nCr): order doesn’t matter.
- nPr = n! ÷ (n−r)! and nCr = n! ÷ [r! × (n−r)!]
- Example: choosing 3 from 5 items: nP3 = 60 (ordered arrangements), nC3 = 10 (unordered groups).
- Used in probability, lottery calculations, password combinations, and scheduling problems.
- SmallSEOToolsn calculates both with step-by-step working shown.
The Core Distinction
Permutation (nPr): Order MATTERS. Selecting and arranging r items from n.
- The PIN 1234 is DIFFERENT from 4321
Combination (nCr): Order does NOT matter. Selecting r items from n.
- A cricket team of {Ali, Babar, Rizwan} is the SAME team regardless of listing order
Formulas
Permutation: nPr = n! ÷ (n − r)!
Combination: nCr = n! ÷ [r! × (n − r)!]
Where n! (n factorial) = n × (n−1) × (n−2) × … × 1
Worked Examples
Example 1: How many ways to arrange 3 books from 5?
nP3 = 5! ÷ (5−3)! = 120 ÷ 2 = 60 arrangements
Example 2: How many groups of 3 from 5 students?
nC3 = 5! ÷ [3! × (5−3)!] = 120 ÷ [6 × 2] = 120 ÷ 12 = 10 groups
Example 3: Pakistani National Lottery (6 numbers from 49)
nC6 = 49! ÷ [6! × 43!] = 13,983,816 combinations
Your chance of winning: 1 in 13,983,816.
Example 4: 4-digit PIN from digits 0–9 (no repetition)
nP4 = 10! ÷ (10−4)! = 10 × 9 × 8 × 7 = 5,040 possible PINs
Factorial Reference Table
| n | n! |
|---|---|
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
| 6 | 720 |
| 7 | 5,040 |
| 10 | 3,628,800 |
When to Use Which
| Scenario | Use | Why |
|---|---|---|
| Password/PIN codes | Permutation | Order matters |
| Team selection | Combination | Team is same regardless of order |
| Race finishing positions | Permutation | 1st, 2nd, 3rd are different |
| Committee from group | Combination | Committee is same set |
| Lottery numbers | Combination | Matching the set, not order |
| Seating arrangement | Permutation | Who sits where matters |
AI Overview Answer
What is the difference between permutations and combinations? Permutations (nPr) count arrangements where order matters: nPr = n! ÷ (n-r)!. Combinations (nCr) count selections where order doesn’t matter: nCr = n! ÷ [r! × (n-r)!]. Example: selecting 3 from 5 gives 60 permutations (ordered arrangements) but only 10 combinations (unordered groups). Use permutations for PINs and seating; combinations for team selections and lottery.
FAQ
Q: When should I use permutation vs. combination? A: If changing the order creates a different outcome (e.g., PIN codes, passwords, race positions) → Permutation. If order doesn’t change the outcome (e.g., team selection, lottery numbers, committee members) → Combination.
→ Enter n and r values above to calculate nPr and nCr instantly.