Combinatorial counting
Choose ordering and replacement explicitly. Counts use bounded BigInt, not rounded floating-point values.
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Choose ordering and replacement explicitly. Counts use bounded BigInt, not rounded floating-point values.
Model, grammar and limits
All variables and coefficients are dimensionless, and angles are in radians. Use explicit multiplication and decimal points. Expression grammar: + - * / ^, parentheses, x (or t, y, v for ODEs; n, a, b for recurrences), pi, e, sin, cos, tan, exp, ln, sqrt and abs. No assignments, code or custom calls.
Expressions: up to 512 characters, 128 tokens and depth 32; generated output up to 8192 characters, 512 tokens and depth 64. Polynomial degree up to 12, matrices up to 6×6 (augmented up to 6×7), 2D or 3D vectors, ODEs up to 2000 steps, Taylor order up to 12, Fourier up to 24 harmonics and 2048 intervals, interpolation up to 50 points (polynomial: up to 12), series up to 2000 terms, recurrences up to 1000 terms with an 8192-bit rational bound. A 15-second worker deadline stops unresolved work.
Exact identities apply only on original domains; numeric residuals, error estimates and sampled plots are not proofs of global accuracy/convergence. This educational model is not a hardware, medical, financial or engineering approval. No input is sent or silently stored; download a raw backup before reset or closing. Input data, formulas and filenames are never sent. Only with your optional consent, Refentra sends aggregate counts of actions and outcomes, without input content.
Formal verification:512 terms,1024 candidate products,32 atoms per monomial,2048 characters per atom,8192-character canonical keys. Rational-root profile: primitive constant/leading integers≤1000000 and20000 candidates; remaining factors are not proved irreducible.
Principal real roots retain their branch: sqrt(x²) is not x for negative x. Gauss–Kronrod 15/7 uses eight initial panels and interval-bounded trigonometric phase changes≤π/2 per panel. Limits:20000 evaluations/depth18. Local error estimates are not guaranteed bounds; unresolved resolution/convergence rejects. Factorial series use expm1 for small amplitudes.
P(n,k), C(n,k), n^k, C(n+k−1,k)
Declared inputs
Declared inputs
| Value | Unit |
|---|---|
| Counting model | Unordered, without replacement |
| Population n 1 | 5 |
| Selection k 1 | 3 |
Sources and licenses
Use the declared profile
- Declared inputs
Choose ordering and replacement explicitly. Counts use bounded BigInt, not rounded floating-point values. All variables and coefficients are dimensionless, and angles are in radians. Use explicit multiplication and decimal points. Expression grammar: + - * / ^, parentheses, x (or t, y, v for ODEs; n, a, b for recurrences), pi, e, sin, cos, tan, exp, ln, sqrt and abs. No assignments, code or custom calls.
- Calculate
P(n,k), C(n,k), n^k, C(n+k−1,k) Expressions: up to 512 characters, 128 tokens and depth 32; generated output up to 8192 characters, 512 tokens and depth 64. Polynomial degree up to 12, matrices up to 6×6 (augmented up to 6×7), 2D or 3D vectors, ODEs up to 2000 steps, Taylor order up to 12, Fourier up to 24 harmonics and 2048 intervals, interpolation up to 50 points (polynomial: up to 12), series up to 2000 terms, recurrences up to 1000 terms with an 8192-bit rational bound. A 15-second worker deadline stops unresolved work.
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Exact identities apply only on original domains; numeric residuals, error estimates and sampled plots are not proofs of global accuracy/convergence. This educational model is not a hardware, medical, financial or engineering approval. No input is sent or silently stored; download a raw backup before reset or closing. Input data, formulas and filenames are never sent. Only with your optional consent, Refentra sends aggregate counts of actions and outcomes, without input content.
Model questions
Which model is used?
Choose ordering and replacement explicitly. Counts use bounded BigInt, not rounded floating-point values.
Which syntax and units are supported?
All variables and coefficients are dimensionless, and angles are in radians. Use explicit multiplication and decimal points. Expression grammar: + - * / ^, parentheses, x (or t, y, v for ODEs; n, a, b for recurrences), pi, e, sin, cos, tan, exp, ln, sqrt and abs. No assignments, code or custom calls.
What do the error estimates and limits mean?
Expressions: up to 512 characters, 128 tokens and depth 32; generated output up to 8192 characters, 512 tokens and depth 64. Polynomial degree up to 12, matrices up to 6×6 (augmented up to 6×7), 2D or 3D vectors, ODEs up to 2000 steps, Taylor order up to 12, Fourier up to 24 harmonics and 2048 intervals, interpolation up to 50 points (polynomial: up to 12), series up to 2000 terms, recurrences up to 1000 terms with an 8192-bit rational bound. A 15-second worker deadline stops unresolved work. Formal verification:512 terms,1024 candidate products,32 atoms per monomial,2048 characters per atom,8192-character canonical keys. Rational-root profile: primitive constant/leading integers≤1000000 and20000 candidates; remaining factors are not proved irreducible.
Where are my inputs processed and saved?
Exact identities apply only on original domains; numeric residuals, error estimates and sampled plots are not proofs of global accuracy/convergence. This educational model is not a hardware, medical, financial or engineering approval. No input is sent or silently stored; download a raw backup before reset or closing. Input data, formulas and filenames are never sent. Only with your optional consent, Refentra sends aggregate counts of actions and outcomes, without input content.