Factorial Calculator — Free Online n! Calculator

How to Use the Factorial Calculator

1. Enter a non-negative integer: Type any whole number from 0 to 1000 into the input field. The calculator only accepts non-negative integers since factorials are not defined for negative numbers or decimals (the gamma function extends to non-integers, but that is a different operation).
2. View the factorial table: For values of 20 or less, a convenient reference table appears on the left showing all factorials from 0! through your entered value. The current value's row is highlighted. This is helpful for quick reference and verification.
3. Read the result: The n! value is displayed prominently. For small values (n ≤ 20), the full multiplication sequence is shown (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120). For larger values, the exact result appears in a scrollable field.
4. Explore properties: The results panel shows the number of digits in the factorial result and the number of trailing zeros. These properties are mathematically interesting and have practical applications in number theory.

The calculator uses JavaScript BigInt for exact arbitrary-precision computation, so every digit of the result is correct even for very large factorials like 500! (which has 1,135 digits). All calculations run locally in your browser.

Factorial Formula and Properties

Variables Explained

• n: A non-negative integer (0, 1, 2, 3, ...) for which the factorial is calculated.
• n!: The factorial of n — the product of all positive integers from 1 to n.
• 0!: Defined as 1 by convention, necessary for combinatorial formulas to work correctly.
• e: Euler's number (approximately 2.71828), used in Stirling's approximation.
• π (pi): Approximately 3.14159, also appears in Stirling's approximation.

Step-by-Step Example

Calculate 7! and find the number of trailing zeros.

1. Write out the multiplication: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1
2. Multiply step by step: 7 × 6 = 42
3. 42 × 5 = 210
4. 210 × 4 = 840
5. 840 × 3 = 2,520
6. 2,520 × 2 = 5,040
7. 5,040 × 1 = 5,040
8. Trailing zeros: ⌊7/5⌋ = 1 (7! = 5,040 has 1 trailing zero)

The factorial 7! = 5,040. It has 4 digits and 1 trailing zero. The trailing zero comes from the single factor of 5 in the product (from 5 itself). The factor of 2 to pair with it comes from any of the even numbers (2, 4, 6).

Practical Examples

Example 1: Rachel's Seating Arrangement Problem

Rachel is organizing a dinner party for 8 guests at a rectangular table. She wants to know how many different seating arrangements are possible.

• For 8 guests in 8 seats, the number of arrangements is 8!
• 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320
• If seating is circular (round table): (8-1)! = 7! = 5,040

There are 40,320 possible seating arrangements for a rectangular table and 5,040 for a round table (circular permutations divide by n because rotations are equivalent). Rachel realizes that trying every arrangement would take far too long and decides to seat people strategically based on conversation compatibility instead.

Example 2: Professor Kim's Combinatorics Lesson

Professor Kim is teaching students about the connection between factorials and combinations. She asks: "In how many ways can you choose 4 books from a shelf of 10?"

• Using the combination formula: C(10, 4) = 10! / (4! × 6!)
• 10! = 3,628,800
• 4! = 24
• 6! = 720
• C(10, 4) = 3,628,800 / (24 × 720) = 3,628,800 / 17,280 = 210

There are 210 ways to choose 4 books from 10. Professor Kim demonstrates the cancellation shortcut: instead of computing the full factorials, calculate (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 5,040 / 24 = 210. This avoids dealing with large intermediate numbers. For more combinatorics problems, use our permutation and combination calculator.

Example 3: Alex's Probability Computation

Alex is computing the probability of dealing a specific 5-card poker hand. He needs to calculate C(52, 5) — the total number of possible 5-card hands from a standard 52-card deck.

• C(52, 5) = 52! / (5! × 47!)
• Simplified: (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1)
• Numerator: 311,875,200
• Denominator: 120
• C(52, 5) = 2,598,960

There are 2,598,960 possible 5-card poker hands. Alex can now calculate specific hand probabilities: a royal flush has 4 possible hands, so P(royal flush) = 4 / 2,598,960 = 0.000154% or about 1 in 649,740. Understanding factorials is essential for these probability calculations.

Factorial Reference Table

Tips and Complete Guide

Efficient Factorial Computation

When using factorials in formulas, always simplify before computing. For C(20, 3) = 20!/(3! × 17!), do not compute 20! then divide by 3! × 17!. Instead, cancel the 17!: (20 × 19 × 18)/(3 × 2 × 1) = 6,840/6 = 1,140. This avoids dealing with astronomically large intermediate numbers and prevents overflow errors. Our calculator handles the full computation, but understanding simplification is essential for manual work and programming.

Factorials in Taylor Series

Factorials appear in the denominators of Taylor series, which are used to approximate functions. The exponential function e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ... . The sine function sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ... . The rapid growth of factorials in the denominator ensures these series converge. This is how calculators and computers actually compute trigonometric and exponential functions internally.

Double Factorials and Subfactorials

Beyond regular factorials, mathematicians use double factorials and subfactorials. The double factorial n!! multiplies every other number: 7!! = 7 × 5 × 3 × 1 = 105, and 8!! = 8 × 6 × 4 × 2 = 384. The subfactorial !n counts derangements (permutations where no element appears in its original position). For example, !3 = 2 (the derangements of 3 are 1 and 2). The formula is !n = n! × Σ(-1)^k/k! for k from 0 to n.

Common Mistakes to Avoid

• Forgetting that 0! = 1: This is not intuitive but is mathematically necessary. Many students assume 0! = 0, which leads to errors in combinatorial formulas. Remember: 0! = 1 by definition.
• Computing full factorials unnecessarily: In expressions like n!/(n-k)!, cancel common terms. P(100, 3) = 100 × 99 × 98 = 970,200. Computing 100! and 97! separately is wasteful and can cause overflow.
• Assuming factorials exist for negatives or decimals: The factorial function n! is only defined for non-negative integers. For non-integer values, use the gamma function: Γ(n+1) = n! for integers, but Γ is defined for all complex numbers except non-positive integers.
• Underestimating factorial growth: Students often underestimate how fast factorials grow. 20! is already about 2.4 × 10^18. An algorithm with O(n!) complexity becomes infeasible beyond n ≈ 15-20 on modern computers.
• Using floating-point for exact counting: Standard floating-point numbers lose precision beyond about 18 significant digits (past 21!). For exact results with large factorials, arbitrary-precision arithmetic (like our BigInt implementation) is essential.

Frequently Asked Questions