Mr Calcu | Effortlessly generate accurate permutation and combination insights to boost your probability and stats analysis.
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Permutation & Combination Calculator Description
Overview
- Permutations: Arrangements where order matters.
- Combinations: Selections where order does not matter.
- Supports both with and without repetition.
Formulas and Derivations
Without Repetition
P(n,r) = \frac{n!}{(n-r)!}
C(n,r) = \frac{n!}{r!(n-r)!}
Pro Tip: For large n, compute factorials iteratively and reuse intermediate results to improve performance.
With Repetition
P_{rep}(n,r) = n^r
C_{rep}(n,r) = \frac{(n+r-1)!}{r!(n-1)!}
Edge Case Considerations
- r = 0: P(n,0)=1 and C(n,0)=1.
- r = n: P(n,n)=n! and C(n,n)=1.
- r > n without repetition: Result = 0.
- r > n with repetition: P_{rep}(n,r)=n^r; C_{rep}(n,r) computed per formula.
- n = 0: Only valid if r=0; P(0,0)=1 and C(0,0)=1; else 0.
- Negative inputs: Invalid; rejects n<0 or r<0.
- Large n or r: Factorials grow rapidly; for n>170, consider Stirling's approximation.
Real-World Case Studies
- Lottery Odds: Odds of selecting 6 numbers from 49:
- C(49,6) = \frac{49!}{6!43!} ≈ 13,983,816.
- Password Strength: 8-character passwords from 62 symbols:
- P_{rep}(62,8) = 62^8 ≈ 218,340,105,584,896.
Start calculating now to unlock precise insights and elevate your analytical capabilities!