Statistics Tool
Combination Calculator
Calculate combinations C(n, r) or binomial coefficients. Learn how to compute selections where order does not matter, with full factorials.
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Interactive Combination Calculator Workspace
Combinations Result (n choose r)210Formula Used: n! / [ r! * (n - r)! ]
Step-by-Step Combinations Analysis
Step 1: Write combination formula: nCr = n! / [ r! × (n - r)! ]
Step 2: Substitute values: 10C4 = 10! / [ 4! × (10 - 4)! ] = 10! / [ 4! × 6! ]
Step 3: Simplify the factorial division: 10 × 9 × 8 × 7 / 4!
Step 4: Solve: 5,040 / 24 = 210
Mathematical Formula & Variables
Equation Model
nCr = n! / [ r! * (n - r)! ]Combinations calculate the number of unique ways to select r items from a set of n items where the order of selection does not matter.
Variable Definitions
| Symbol | Description |
|---|---|
| n | Total number of items in the set |
| r | The number of items to choose |
| nCr | Total number of unique unordered combinations (also written as n choose r) |
How to Use the Combination Calculator
- Enter the total number of items (n).
- Enter the number of items to choose (r).
- Toggle "Allow Repetition" if needed (calculated as (n+r-1)! / (r! * (n-1)!)).
- Analyze the division of factorials and the final count.
Practical Example Calculation
Scenario Context: Choosing a 3-person volunteer group (r=3) from a team of 10 people (n=10).
Step 1: Write the formula: 10C3 = 10! / [ 3! * (10 - 3)! ] = 10! / (3! * 7!).
Step 2: Expand factorials: 10! = 10 * 9 * 8 * 7!.
Step 3: Simplify: (10 * 9 * 8) / 3! = 720 / 6 = 120.
There are 120 unique ways to form the volunteer group.
Frequently Asked Questions
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