Permutations and combinations are the math of counting arrangements — how many ways you can pick, arrange, or order items from a set. They show up in lottery odds, password security, scheduling, and dozens of competitive exam questions (JEE, CAT, GMAT, government exams).
This guide explains the critical difference between permutations and combinations, the formulas, and worked examples for the typical question types you'll encounter.
Calculate Permutations & Combinations — Free
P(n,r), C(n,r), factorials with step-by-step solutions. Perfect for JEE, CAT, GMAT prep.
Permutation vs Combination
| Aspect | Permutation | Combination |
|---|---|---|
| Order matters? | Yes | No |
| Notation | P(n,r) or nPr | C(n,r) or nCr |
| Formula | n! / (n-r)! | n! / [r!(n-r)!] |
| Example | Race finishing positions | Selecting team members |
| 3 from {A,B,C} arranged in 2 spots | AB, BA, AC, CA, BC, CB = 6 | AB, AC, BC = 3 |
Ask "Does ABC differ from CBA in this problem?" Yes = permutation. No = combination.
Formulas Explained
Permutation: P(n,r) = n! / (n-r)!
Number of ways to arrange r items from n distinct items.
Combination: C(n,r) = n! / [r!(n-r)!]
Number of ways to choose r items from n, where order doesn't matter.
Factorial
n! = n × (n-1) × (n-2) × ... × 2 × 1
5! = 5 × 4 × 3 × 2 × 1 = 120
0! = 1 (by definition)Worked Examples
Example 1: Permutation
How many 3-letter codes from 26 letters (no repeat)?
P(26,3) = 26! / (26-3)! = 26 × 25 × 24 = 15,600Example 2: Combination
How many ways to pick 5 cricketers from 11?
C(11,5) = 11! / (5! × 6!) = 462Example 3: With Repetition Allowed
How many 4-digit PINs (digits 0-9, can repeat)?
10 × 10 × 10 × 10 = 10,000 (NOT permutation since repetition allowed)Example 4: Lottery Odds
Pick 6 numbers from 50, order doesn't matter:
C(50,6) = 15,890,700 — your odds of winningCommon P&C Mistakes
- Confusing P and C — always check if order matters
- Forgetting "no repetition" rule — different from "repetition allowed"
- Misreading "selection" as "arrangement"
- Adding instead of multiplying for stages — use multiplication principle
- Treating identical objects as distinct — affects formula
How to Use the Tool (Step by Step)
- 1
Enter n (total items)
Total available items in the set.
- 2
Enter r (selected items)
How many you're picking/arranging.
- 3
Pick Operation
Permutation P(n,r), Combination C(n,r), or Factorial n!.
- 4
See Result
Tool computes value with step-by-step calculation.
- 5
Verify with Manual
For small numbers, double-check by manual enumeration.
Frequently Asked Questions
When should I use permutation vs combination?+−
Use permutation when order matters (PIN, race finishers). Use combination when order doesn't matter (selecting a team, lottery numbers).
What is 0!?+−
0! = 1 by convention. This makes formulas work consistently. C(n,0) = 1 (one way to choose nothing).
Can n and r be equal in P(n,r)?+−
Yes. P(n,n) = n! — total arrangements of all n items.
What is the difference between P(n,r) and n^r?+−
P(n,r) is for no repetition. n^r is for repetition allowed. PIN of 4 digits = 10^4 = 10,000 (repetition allowed).
How are P&C used in real life?+−
Password strength, lottery odds, route planning, scheduling, statistics, cryptography, sports tournament fixtures.
Are large factorials computable?+−
Up to 170! for double-precision floats. Beyond that, use logarithms or BigInt arithmetic. Most calculators handle up to 169!.
Calculate Permutations & Combinations — Free
P(n,r), C(n,r), factorials with step-by-step solutions. Perfect for JEE, CAT, GMAT prep.
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