🧮 Combinations & Binomial Coefficients
Compute combinations for large n and r, distribute identical items with stars-and-bars, and gauge lottery odds. The calculator uses BigInt for exact results when possible and a log-gamma approximation for extremely large values.
1. Standard Combination (n choose r)
Order does not matter, selections without replacement.
Combinations
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2. Stars & Bars (Identical Items)
Distribute identical items into distinct bins (non-negative counts).
Ways to distribute
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3. Lottery Odds (n choose r)
Estimate the chance of hitting a jackpot from a single ticket.
Total combinations
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How to Use
- Select the calculator panel that matches your scenario: standard nCr, stars-and-bars, or lottery odds.
- Enter non-negative integers respecting the constraints (e.g., 0 ≤ r ≤ n for nCr, at least one bin for stars-and-bars).
- Press the relevant compute button to view exact or scientific-notation outputs. Messages will guide you if inputs are inconsistent.
- Reset individual sections when starting a fresh calculation to avoid accidental reuse of previous values.
Formula References
- Wilf, H. S. (1990). Generatingfunctionology. Academic Press.
- NIST Digital Library of Mathematical Functions. “Combinatorial Identities.” https://dlmf.nist.gov