Combination Calculator
Calculate nCr combinations for unordered item subsets.
GCD & LCM Calculator
About Combination Calculator
The Combination Calculator ($nCr$ / Binomial Coefficient) computes the number of unique unordered subsets of $r$ items selected from a set of $n$ distinct elements with and without repetition, displaying Pascal's Triangle coordinates and factorial cancellations.
How to Use Combination Calculator
Step 1
Enter total items in set ($n$) and number of items to choose ($r$).
Step 2
Click "Calculate Combination".
Step 3
Inspect the factorial formula steps and binomial coefficient.
Step 4
Click "Copy Result".
Practical Use Cases for Combination Calculator
Lottery & Poker Hand Combinatorics
Calculate the total number of possible 5-card poker hands from a 52-card deck ($_{52}C_5 = 2,598,960$) and lottery jackpot odds.
Team Formation & Committee Selection
Determine how many ways to select a 4-person committee from a department of 15 employees ($_{15}C_4 = 1,365$).
Input & Output Examples
Selecting 4 Team Members from 10 Candidates
Total elements (n): 10, Subset size (r): 4
$_{10}C_4 = \\frac{10!}{4!(10-4)!} = \\frac{3628800}{24 \\times 720} = 210$ combinationsKey Features & Performance
- ✓Calculates $nCr = \\binom{n}{r} = \\frac{n!}{r!(n-r)!}$ (Without Repetition) and $\\binom{n+r-1}{r}$ (With Repetition).
- ✓Displays Pascal's Triangle row coordinates and step-by-step multiplication.
- ✓BigInt arbitrary-precision arithmetic guarantees exact integer calculations.
- ✓100% Client-Side calculation.
- ✓1-Click Copy combination solution.
Key Terminology & Definitions
Combination ($nCr$ / $\\binom{n}{r}$)
A selection of items from a collection where the order of selection does not matter: $\\binom{n}{r} = \\frac{n!}{r!(n-r)!}$.
Binomial Coefficient Symmetry
$\\binom{n}{r} = \\binom{n}{n-r}$, meaning choosing $r$ items to include is mathematically identical to choosing $n-r$ items to leave out.