Combination Calculator
Calculate the total number of unordered combinations (nCr) of choosing r elements from a set of n items.
Input Parameters
Live Result
Details
Steps
State Newton binomial combination formula
Simplify factorial terms in numerator and denominator
Obtain final unordered combination count
Quick Answer & Understanding Your Result
A Combination Calculator computes unordered subsets (nCr or n choose r), calculating how many ways you can select r items from a set of n elements where order does not matter.
Core Combinatorial Principles
- Order Does Not Matter (Subsets): Grouping {A, B} is identical to {B, A} (nCr = n! / (r! × (n - r)!)).
- Binomial Coefficient Connection: The value nCr corresponds directly to the coefficients in Pascal's Triangle and Binomial Expansions ((a + b)ⁿ).
- Symmetry Property: nCr = nC(n - r) (e.g. choosing 2 from 10 is identical to leaving 8 behind: 10C2 = 10C8 = 45).
Probability & Everyday Applications
- Lottery & Card Hands: Calculating poker hand probabilities (e.g. 5 cards from 52).
- Committee & Team Formation: Selecting delegation teams from a pool of candidates.
Enter total elements (n) and subset size (r) above to calculate combinations and step-by-step solutions instantly.
Formula & Calculation Logic
Mathematical FormulaBinomial formula:
- Divides permutation count nPr by r! to eliminate duplicate orderings of identical subsets.
Step-by-Step Guide
Enter Total Universe (n)
Input the size of the pool (e.g. 52 cards, 10 team members).
Enter Selection Size (r)
Input how many items to choose (e.g. 5 cards, 3 members).
View Combinations & Steps
Receive total unique subset combinations and binomial coefficient breakdown.
Worked Examples Table
| Example | Input Values | Calculated Result | Explanation |
|---|---|---|---|
| Poker Hand: 5 Cards from 52 (52C5) | 52! ÷ (5! × 47!) = 2,598,960. | ||
| Poker Hand (5 cards from 52, C(52, 5) = 2,598,960) | Provides instant quantitative evaluation for comparative practical analysis. |
Frequently Asked Questions
Evidence-Based Methodology & Authoritative References
NIST Digital Library of Mathematical Functions - Binomial Coefficients
Khan Academy - Combinations and Probability
Standard: ISO / W3C / IETF Authoritative Domain Standard
Formula: nCr = n! ÷ (r! × (n - r)!)
Last Reviewed: August 21, 2026
Primary Source: NIST Digital Library of Mathematical Functions - Binomial Coefficients
Reference Document: Khan Academy - Combinations and Probability
Computation Privacy: 100% client-side 0ms computation, zero data retention
Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.
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Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.