Factorial Calculator

Calculate n! (n Factorial) for any non-negative integer | Step-by-step solution included

🔢 Enter a non-negative integer (0–170)

💡 n must be a whole number between 0 and 170

Result

📝 Step-by-step solution

Enter a number and click "Calculate n!"

💡 n! = n × (n−1) × (n−2) × ... × 2 × 1 | 0! = 1 by definition

📖 What is a Factorial?

The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. It is a fundamental concept in combinatorics, probability, and calculus. Factorials count the number of ways to arrange n distinct objects in a sequence — also known as permutations.

n! = n × (n − 1) × (n − 2) × ... × 2 × 1

Special case: 0! = 1 (by convention)

📝 Worked Example

Example: Calculate 5!

Step 1: Write the product: 5! = 5 × 4 × 3 × 2 × 1
Step 2: Multiply step by step:
    5 × 4 = 20
    20 × 3 = 60
    60 × 2 = 120
    120 × 1 = 120
Result: 5! = 120

📊 Factorial Table (Common Values)

n	n!	Value
0	0!	1
1	1!	1
2	2!	2
3	3!	6
4	4!	24
5	5!	120
6	6!	720
7	7!	5,040
8	8!	40,320
9	9!	362,880
10	10!	3,628,800
15	15!	1,307,674,368,000
20	20!	2.4329 × 10¹⁸
50	50!	3.0414 × 10⁶⁴
100	100!	9.3326 × 10¹⁵⁷

🎯 Real-World Applications of Factorials

Combinatorics: Count arrangements and combinations (nCr, nPr).
Probability: Calculate odds in card games, lotteries, and dice rolls.
Calculus: Taylor series and Maclaurin series use factorials.
Computer Science: Recursion, dynamic programming, algorithm complexity.
Statistics: Binomial distribution and Poisson distribution.
Cryptography: Factorials appear in some encryption algorithms.

🧮 Useful Factorial Properties

Recurrence: n! = n × (n − 1)!
Stirling's approximation: n! ≈ √(2πn) × (n/e)ⁿ
Ratio: n! / (n − 1)! = n
Relationship to combinations: nCr = n! / (r! × (n − r)!)

⚠️ Limitations & Assumptions

📌 Important Notes

This calculator uses JavaScript numbers (double precision), which lose accuracy above n = 170.
For n > 170, the result overflows to Infinity in JavaScript.
For very large n, results are shown in scientific notation (e.g., 1.234e+50).
Only non-negative integers are supported. Fractions and negative numbers are not valid in standard factorials.
Gamma function extends factorials to real numbers, but this is not implemented here.

📚 References & Sources

Authoritative Sources

Weisstein, Eric W. — Factorial, MathWorld
NIST Digital Library of Mathematical Functions — Gamma Function
Khan Academy — Factorials in Combinatorics
Abramowitz & Stegun — Handbook of Mathematical Functions

❓ Frequently Asked Questions

What is 0! (zero factorial)?

By definition, 0! = 1. This is not arbitrary — it makes the recurrence n! = n × (n − 1)! work for n = 1: 1! = 1 × 0!, so 0! must equal 1.

Can factorials be calculated for negative numbers?

No — standard factorials are only defined for non-negative integers. The Gamma function extends factorials to real numbers, but negative integers remain undefined.

Why does 170! overflow?

170! is approximately 7.26 × 10³⁰⁶, which is very close to the maximum double-precision floating point value (~1.8 × 10³⁰⁸). 171! exceeds that limit.

What is the relationship between factorials and permutations?

The number of permutations of n distinct objects is exactly n!. For example, there are 5! = 120 ways to arrange 5 different books on a shelf.

How large is 100!?

100! ≈ 9.3326 × 10¹⁵⁷ — a 158-digit number, larger than the number of atoms in the observable universe (approximately 10⁸⁰).

🔗 Related Calculators

⚠️ Disclaimer

This Factorial Calculator is provided for educational and informational purposes only. Results are computed using JavaScript double-precision floating point arithmetic, which is accurate up to approximately 170!. For very large factorials or precision-critical applications, use dedicated mathematical software such as Wolfram Mathematica, Python's math.factorial, or similar tools. Online Calculator Zone is not responsible for any decisions made based on these calculations.