Calculator Inputs
Example Data Table
| Case | Expression | Simplified value | Method |
|---|---|---|---|
| Ratio | 8! / 5! | 8 × 7 × 6 = 336 | Cancel 5! |
| Permutation | 6P3 | 6! / 3! = 120 | Keep three descending terms |
| Combination | 10C4 | 210 | Divide by repeated arrangements |
| Custom | 10!4! / 7!3! | 960 | Use prime exponent cancellation |
Formula Used
Factorial: n! = n × (n - 1) × ... × 2 × 1, with 0! = 1.
Ratio: n! / r! cancels the common lower factorial. If n ≥ r, the result is (r + 1) × ... × n.
Permutation: nPr = n! / (n - r)!. It counts ordered selections.
Combination: nCr = n! / [r!(n - r)!]. It counts unordered selections.
Prime exponent method: each factorial is decomposed into prime powers. Denominator exponents are subtracted from numerator exponents.
How to Use This Calculator
Choose the simplification type first. Enter n and r for standard factorial ratios, permutations, or combinations. Use custom numerator and denominator lists when your expression has several factorials. Press the main button. The simplified result appears above the form. Review exact value, scientific notation, prime factor form, and steps. Use CSV or PDF buttons to save the result.
Factorial Simplification Guide
What factorial simplification means
Factorial simplification reduces large products before direct multiplication. It is useful in algebra, probability, statistics, and counting problems. A factorial grows very fast. Direct expansion can become long and hard to check. Simplification cancels shared terms first. Then the final expression becomes smaller and clearer.
Why cancellation matters
Many factorial expressions contain a larger factorial divided by a smaller factorial. For example, 9! divided by 6! does not require writing every factor. The 6! portion cancels. Only 9 × 8 × 7 remains. This method saves time and reduces arithmetic errors. It also shows the structure behind permutations and combinations.
Prime factor simplification
Advanced expressions may include several factorials in the numerator and denominator. Simple cancellation may not be obvious. Prime factorization solves this problem. Each factorial is changed into powers of primes. The denominator powers are subtracted from numerator powers. Any positive powers stay on top. Any negative powers stay below. This produces a compact exact form.
Exact and approximate results
Small inputs can be printed as exact whole numbers. Large inputs may create hundreds or thousands of digits. In those cases, scientific notation is easier to read. The calculator also estimates digit length. That helps you understand the result size without filling the page with huge numbers.
Best use cases
Use this calculator for homework checks, probability formulas, binomial coefficients, arrangements, selections, and expression cleanup. It is also useful when comparing two factorial expressions. Enter values carefully. Check that r is not greater than n in permutations and combinations. For custom mode, place factorial values from the top expression in the numerator field. Place lower expression values in the denominator field.
Common mistakes to avoid
Do not expand a huge factorial too early. Cancel first, then multiply. Watch the order of numerator and denominator entries. A reversed ratio gives a different result. Keep r within the allowed range for nPr and nCr. Use custom mode only for factorial numbers, not ordinary multiplied numbers. When a result is fractional, study the denominator factors. They show which parts could not cancel from the original expression. This habit improves accuracy and keeps every step easier to explain during review.
FAQs
What is a factorial?
A factorial multiplies a positive integer by every smaller positive integer. For example, 5! equals 5 × 4 × 3 × 2 × 1, which is 120.
Why is 0! equal to 1?
Zero factorial is defined as 1 because it keeps counting formulas consistent. It represents one way to arrange or choose nothing.
How does factorial cancellation work?
Shared factorial parts cancel first. In 8! / 5!, the 5! terms vanish, leaving 8 × 7 × 6.
Can this calculator handle combinations?
Yes. Select combination mode, then enter n and r. It applies n! divided by r! times (n - r)!.
Can this calculator handle permutations?
Yes. Select permutation mode. The calculator uses n! divided by (n - r)! and returns ordered arrangements.
What does custom mode do?
Custom mode simplifies products of factorials over other products of factorials. Enter top factorial numbers and bottom factorial numbers separately.
Why use prime factors?
Prime factors make complex cancellation reliable. They show exactly which factors remain after numerator and denominator powers are compared.
Why is scientific notation shown?
Factorial results grow quickly. Scientific notation gives a readable size estimate when the exact number becomes very long.