What is Stirling's approximation for factorials?

Stirling's approximation estimates large factorials: n! ≈ √(2πn) × (n/e)^n. This formula becomes more accurate as n increases and is useful when exact factorial values are impractical to compute. For n = 10, it gives 3,598,695 compared to the exact 3,628,800 - about 99.2% accurate.

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Q: How do you count trailing zeros in a factorial?

Trailing zeros in n! come from factors of 10, which require pairs of 2 and 5. Since there are always more factors of 2 than 5, count the factors of 5. The formula is: floor(n/5) + floor(n/25) + floor(n/125) + ... For example, 100! has floor(100/5) + floor(100/25) + floor(100/125) = 20 + 4 + 0 = 24 trailing zeros.

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About Factorial Calculator

The Factorial Calculator computes the factorial of any non-negative integer n, written as n! (pronounced "n factorial"). Factorial is the product of all positive integers from 1 to n, and this tool supports calculations for values as large as one million, displaying results in both full form and scientific notation.

What is Factorial?

The factorial of a non-negative integer n is the product of all positive integers less than or equal to n. It is denoted by n! and defined as:

Factorial Definition

$$n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1$$

By convention, 0! is defined as 1. This is not arbitrary - it ensures that many mathematical formulas work correctly and maintains the recursive relationship n! = n × (n-1)!

Examples of Factorials

0! = 1 (by definition)
1! = 1
5! = 5 × 4 × 3 × 2 × 1 = 120
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

How to Use This Calculator

Enter your number: Type any non-negative integer from 0 to 1,000,000 in the input field, or use the quick-select buttons for common values.
Click Calculate: Press the "Calculate Factorial" button to compute n!.
View your result: See the factorial value, expansion formula, digit count, and trailing zeros analysis.
Review step-by-step: For small values (≤12), view the complete multiplication breakdown.

Understanding Your Results

Full Result: The complete factorial value (shown for n ≤ 9999)
Scientific Notation: For large results, displayed as mantissa × 10^exponent
Digit Count: How many digits are in the factorial result
Trailing Zeros: How many zeros the result ends with
Expansion: The multiplication formula n × (n-1) × ... × 1

Applications of Factorials

🎲 Permutations

Calculate the number of ways to arrange n distinct objects. For example, 5 books can be arranged on a shelf in 5! = 120 different ways.

🎯 Combinations

Find how many ways to choose k items from n items using the formula C(n,k) = n! / (k!(n-k)!), fundamental in probability theory.

📐 Binomial Theorem

Factorials appear in binomial coefficients used to expand expressions like (a+b)^n in algebra and calculus.

∑ Taylor Series

Many important functions are expressed as infinite series involving factorials, such as e^x = Σ(x^n/n!) and sin(x).

The Growth of Factorials

Factorials grow at a super-exponential rate - faster than any exponential function. This rapid growth is why factorials are important in complexity theory and algorithm analysis.

n	n!	Digits	Trailing Zeros
5	120	3	1
10	3,628,800	7	2
20	2,432,902,008,176,640,000	19	4
50	≈ 3.04 × 10^64	65	12
100	≈ 9.33 × 10^157	158	24
1000	≈ 4.02 × 10^2567	2,568	249

Why is 0! = 1?

The definition 0! = 1 is a mathematical convention that makes many formulas work correctly:

Recursion: The relation n! = n × (n-1)! implies 1! = 1 × 0!, so 0! must equal 1.
Combinatorics: There is exactly one way to arrange zero objects - by doing nothing.
Gamma function: The generalized factorial Γ(1) = 0! = 1.
Empty product: The product of no numbers is defined as 1 (the multiplicative identity).

Trailing Zeros in Factorials

The number of trailing zeros in n! equals the number of times 10 divides n!. Since 10 = 2 × 5 and there are always more factors of 2 than 5, we count factors of 5:

Trailing Zeros Formula

$$\text{Trailing zeros} = \left\lfloor\frac{n}{5}\right\rfloor + \left\lfloor\frac{n}{25}\right\rfloor + \left\lfloor\frac{n}{125}\right\rfloor + \ldots$$

Stirling's Approximation

For large n, calculating n! exactly becomes impractical. Stirling's approximation provides an estimate:

Stirling's Approximation

$$n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$$

This approximation becomes increasingly accurate as n grows larger and is useful for theoretical calculations.

Frequently Asked Questions

What is a factorial?

A factorial, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials grow extremely rapidly - 20! already has 19 digits, and 100! has 158 digits.

Why is 0 factorial equal to 1?

0! = 1 by mathematical convention. This definition makes many mathematical formulas work correctly, particularly in combinatorics where the number of ways to arrange zero objects (an empty set) is one way - by doing nothing. It also maintains the recursive property that n! = n × (n-1)!.

How fast do factorials grow?

Factorials grow faster than exponential functions. While 10! = 3,628,800, just 20! exceeds 2 quintillion. 100! has 158 digits, and 1000! has 2,568 digits. This super-exponential growth is why factorials appear in complexity theory.

What are factorials used for?

Factorials are fundamental in combinatorics for counting permutations and combinations. They appear in probability theory, the binomial theorem, Taylor series expansions, and are essential in statistics, physics, and computer science.

How do you count trailing zeros in a factorial?

Trailing zeros come from factors of 10 (= 2 × 5). Count factors of 5 since there are always more factors of 2. Use: floor(n/5) + floor(n/25) + floor(n/125) + ... For example, 100! has 20 + 4 + 0 = 24 trailing zeros.

What is Stirling's approximation?

Stirling's approximation estimates large factorials: n! ≈ √(2πn) × (n/e)^n. It becomes more accurate as n increases and is useful when exact values are impractical to compute.

Additional Resources

Reference this content, page, or tool as:

"Factorial Calculator" at https://MiniWebtool.com/factorial-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 18, 2026

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Factorial Calculator

About Factorial Calculator

What is Factorial?

Examples of Factorials

How to Use This Calculator

Understanding Your Results

Applications of Factorials

🎲 Permutations

🎯 Combinations

📐 Binomial Theorem

∑ Taylor Series

The Growth of Factorials

Why is 0! = 1?

Trailing Zeros in Factorials

Stirling's Approximation

Frequently Asked Questions

What is a factorial?

Why is 0 factorial equal to 1?

How fast do factorials grow?

What are factorials used for?

How do you count trailing zeros in a factorial?

What is Stirling's approximation?

Additional Resources

Advanced Math Operations:

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