PERMUTATION & COMBINATION CALCULATOR
Order Matters ($nPr$) $\cdot$ Order Doesn't Matter ($nCr$)
Probability & Combinatorics
Results
Permutations ($nPr$)
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Combinations ($nCr$)
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Factorial ($n!$)
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Factorial ($r!$)
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Step-by-Step Breakdown
Permutation & Combination Calculator
In combinatorics and probability theory, Permutations and Combinations determine the total number of unique ways to select a subset of items from a larger set. The primary distinction between the two lies in whether the order of selection matters.
1. Mathematical Definitions & Formulas
| Concept | Order Matters? | Formula | Example Scenario |
|---|---|---|---|
| Permutation ($nPr$) | Yes | $$P(n, r) = \frac{n!}{(n - r)!}$$ | Lock combinations, race finishing orders, assigned seating. |
| Combination ($nCr$) | No | $$C(n, r) = \frac{n!}{r!(n - r)!}$$ | Lottery numbers, team selection, hand of cards. |
2. Key Properties of Combinatorics
- Symmetry of Combinations: Selection of $r$ items is identical to leaving $n-r$ items behind: $$\binom{n}{r} = \binom{n}{n-r}$$
- Relationship Between $nPr$ and $nCr$: Permutations can be calculated directly by multiplying combinations by the number of arrangements ($r!$): $$nPr = nCr \times r!$$
- Zero Factorial Definition: By mathematical definition, $0! = 1$, ensuring calculations for picking all $n$ items ($\binom{n}{n} = 1$) or zero items ($\binom{n}{0} = 1$) remain consistent.
Frequently Asked Questions
What is the difference between $nPr$ and $nCr$?
Use Permutation ($nPr$) when the sequence or order of chosen items matters (e.g., arrange $A, B, C$ vs $B, A, C$). Use Combination ($nCr$) when group membership is all that matters, regardless of order.
Why is $n = 170$ the maximum limit in standard JavaScript calculators?
JavaScript uses double-precision 64-bit floating-point arithmetic. $170! \approx 7.25 \times 10^{306}$, which is close to the maximum finite number representable ($1.79 \times 10^{308}$). Factorials for numbers greater than $170$ evaluate to Infinity.
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