Factorial

1. Enter a non-negative integer value for **n** in the input field (e.g., 5, 10, 20).
2. Click the **Calculate** button (or press Enter) to compute the result.
3. Read the exact value of **n!** displayed in the result area.
4. For large values of n (e.g., n = 100), note that the result is an extremely large integer — the calculator displays the full exact value.
5. Use the **Reset** button to clear the input and start a new calculation.

n! = n × (n−1) × (n−2) × … × 2 × 1,   with 0! = 1

• n — The non-negative integer whose factorial you want to compute. Must be a whole number ≥ 0.
• n! — The factorial of n — the product of all positive integers from 1 to n. For n = 0, the result is 1 by definition.
• (n−1)! — The factorial of the previous integer, used in the recursive definition: n! = n × (n−1)!

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

Result: 7! = 5,040

The result 5,040 means there are 5,040 different ways to arrange 7 distinct objects in a sequence. This is the foundational use of factorials in permutations: the number of ways to order n items is always n!. Factorials grow extremely fast — for reference, 10! = 3,628,800 and 20! = 2,432,902,008,176,640,000.

What Is a Factorial?

A factorial (denoted n!) is one of the most important operations in combinatorics, probability, and analysis. It represents the total number of ways to arrange n distinct objects in order.

Key Properties

• 0! = 1 — defined by convention; consistent with the empty product and required for combinatorial formulas to work at n = 0.
• 1! = 1
• Recursive property: n! = n × (n−1)!
• Factorial growth is super-exponential — values become astronomically large very quickly.

Where Are Factorials Used?

| Application | Formula | |---|---| | Permutations (ordered arrangements of n items) | n! | | Permutations of k items from n | n! / (n−k)! | | Combinations (binomial coefficients) | n! / (k! × (n−k)!) | | Probability of a specific ordering | 1 / n! | | Taylor series (e.g., e^x, sin x) | Denominators involve n! |

Factorial Growth Table

| n | n! | |---|---| | 0 | 1 | | 1 | 1 | | 2 | 2 | | 3 | 6 | | 4 | 24 | | 5 | 120 | | 6 | 720 | | 7 | 5,040 | | 10 | 3,628,800 | | 15 | 1,307,674,368,000 | | 20 | 2,432,902,008,176,640,000 |

The Gamma Function Extension

Factorials are defined only for non-negative integers, but mathematicians extended the concept to real and complex numbers using the Gamma function: Γ(n) = (n−1)!, so Γ(n+1) = n! for positive integers.

• **Using negative integers:** Factorials are undefined for negative integers. Only n ≥ 0 is valid.
• **Confusing n! with n×n:** n! is NOT the same as n squared or n times itself — it is the product of all integers from 1 to n.
• **Forgetting 0! = 1:** A common error is assuming 0! = 0. By mathematical convention and the empty product rule, 0! = 1.
• **Misapplying in combinations:** In the formula C(n,k) = n! / (k!(n-k)!), forgetting to divide by both k! and (n−k)! is a frequent mistake.
• **Overflow in manual calculations:** For large n (e.g., n ≥ 13), factorials exceed typical 32-bit integer limits. Use this calculator for accurate large-number results.

• Factorial — Wolfram MathWorld — Wolfram Research
• OEIS A000142: Factorial numbers — The On-Line Encyclopedia of Integer Sequences (OEIS)
• Concrete Mathematics: A Foundation for Computer Science (Graham, Knuth, Patashnik) — Pearson / Addison-Wesley
• NIST Digital Library of Mathematical Functions — Gamma Function (§5.2) — National Institute of Standards and Technology (NIST)

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