Combinations (nCr) & Permutations (nPr) Calculator
Compute combinations (n choose r) and permutations (n pick r) without overflow.
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How to Use the Combination Calculator: nCr & nPr Permutations
- 1Enter the total number of items in your set (n).
- 2Enter the number of items chosen or selected in each sample (r).
- 3Review the total number of combinations (nCr) where arrangement order is irrelevant.
- 4Compare with permutations (nPr) where item sequencing order is strictly distinguished.
Formula & How It Works
Factorial Combinatorics Equations
Calculates arrangements using factorials and multiplicative cancellations.
Worked Examples
Choosing 3 winners from 10 participants
Input: n = 10, r = 3
Explanation: If prizes are identical, there are 120 combinations; if prizes are 1st, 2nd, and 3rd place, there are 720 permutations.
Frequently Asked Questions
What is the key difference between combinations and permutations?▼
In combinations, order does not matter (e.g. hand of cards); in permutations, order matters (e.g. combination lock PIN code).
Why is a combination lock actually a permutation lock?▼
Because the exact order of entering numbers matters (1-2-3 is different from 3-2-1), which mathematically defines a permutation.
What is the value of 0 factorial (0!)?▼
By mathematical definition, 0! equals 1.
What is the maximum value of n supported?▼
This tool supports standard combinatorial calculations up to n = 100 with accurate cancellation algorithms.
Quick Reference & Conversion Table
| Total Items (n) | Chosen (r) | Combinations nCr | Permutations nPr |
|---|---|---|---|
| 5 | 2 | 10 | 20 |
| 6 | 3 | 20 | 120 |
| 10 | 2 | 45 | 90 |
| 52 (Cards) | 5 (Poker) | 2,598,960 | 311,875,200 |
⚠️ Common Pitfalls to Avoid
- •Confusing combinations (order does not matter) with permutations (order matters).
- •Entering sample size r greater than total population n.
💡 Practical Real-World Uses
- •Calculating winning jackpot odds in state and national lottery drawings.
- •Determining poker hand probabilities and card game draw combinations.
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