Factorial Calculator
Calculate n! for any non-negative integer, including very large factorials, plus permutations (nPr) and combinations (nCr) — with full step-by-step solutions.
Factorial values for n = 0 to 15:
| n | n! |
|---|
f Formulas Used
? How To Use
★ Key Features
Large Number Support
Handles factorials up to 1000! using BigInt precision.
Permutations & Combinations
nPr and nCr calculated together from the same inputs.
Step-by-Step Working
Full multiplication chains shown for smaller factorials.
Mobile Optimized
Clean layout perfect for statistics and probability students.
✎ Real-World Example
49C6 = 49! / (6! × 43!)
= (49×48×47×46×45×44) / (6×5×4×3×2×1)
= 10,068,347,520 / 720
Q Frequently Asked Questions
Factorial Calculator - Calculate n! for Any Number Instantly
Factorials are one of those mathematical concepts that seem simple at first but quickly become incredibly powerful. Multiplying a few numbers together might not sound impressive, yet factorials sit at the heart of probability, statistics, combinatorics, computer science, and countless real-world calculations. A Factorial Calculator makes these calculations instant, allowing you to work with very large values that would otherwise take considerable time to compute by hand.
Whether you're solving a probability problem, calculating permutations and combinations, analyzing algorithms, or preparing for a mathematics exam, understanding factorials can make many complex formulas much easier to understand. This calculator instantly computes n!, displays the multiplication steps, and helps students learn how factorial calculations work behind the scenes.
At first glance, factorial notation may look unusual because of the exclamation mark. However, the idea itself is straightforward. A factorial simply means multiplying a number by every positive whole number below it until you reach 1.
What Is a Factorial?
A factorial is represented using the exclamation mark symbol (!). The factorial of a number n is written as n!.
The formula is:
n! = n × (n − 1) × (n − 2) × ... × 3 × 2 × 1
For example:
5! = 5 × 4 × 3 × 2 × 1 = 120
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
As numbers become larger, factorials grow at an astonishing rate. This rapid growth is why a dedicated calculator becomes so useful.
For example:
10! = 3,628,800
15! = 1,307,674,368,000
20! = 2,432,902,008,176,640,000
Trying to calculate these values manually can be time-consuming and prone to mistakes, which is why many students and professionals rely on a Factorial Calculator.
Why Factorials Matter
Factorials appear throughout mathematics because they count arrangements and possibilities. Whenever you're asking, "How many different ways can this happen?" factorials are often involved.
Imagine arranging five different books on a shelf. The first position can be filled by any of the five books. After choosing one, only four remain for the second position. Then three remain, then two, then one.
The total number of arrangements becomes:
5 × 4 × 3 × 2 × 1 = 120
This is exactly 5!.
Factorials help answer questions involving ordering, ranking, scheduling, seating arrangements, password generation, tournament outcomes, and many other real-world situations.
How to Use This Factorial Calculator
This calculator is designed to be simple enough for beginners while still providing detailed results for advanced users.
Enter any non-negative integer into the calculator and click calculate. The tool instantly computes the factorial value and can also display the multiplication chain used to reach the answer.
For example, entering 8 produces:
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320
This step-by-step breakdown helps students verify homework answers and understand the underlying process rather than simply receiving the final number.
Factorial Calculator — Rules
| Rule | Description |
|---|---|
| Base case | 0! = 1 (by definition) |
| Recursive form | n! = n × (n−1)! |
| Only whole numbers | Factorials are defined only for non-negative integers |
| Grows fast | Factorials increase extremely quickly as n grows |
Real-Life Example: Race Results
Suppose six runners compete in a race. How many different finishing orders are possible?
The first position can be occupied by any of the six runners.
After the winner is chosen, five runners remain for second place.
Then four runners remain for third place, and so on.
The total number of possible finishing orders becomes:
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
Therefore, six runners can finish in 720 different orders.
This type of calculation appears frequently in probability and statistics, where factorials form the foundation of many formulas.
Understanding Factorial Growth
One of the most fascinating aspects of factorials is how quickly they increase.
Consider the following sequence:
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5,040
8! = 40,320
9! = 362,880
10! = 3,628,800
Notice how each step multiplies the previous factorial by the next number. This causes factorials to grow much faster than ordinary multiplication, powers, or even many exponential functions.
If you're comparing factorial growth with powers and exponents, our Exponent Calculator provides a useful comparison between these different mathematical operations.
The Special Case of 0!
Many students are surprised to discover that:
0! = 1
At first this seems strange. After all, how can multiplying nothing produce 1?
The answer comes from mathematical consistency. Many formulas involving permutations and combinations only work correctly if 0! equals 1.
Another way to understand it is through arrangements. There is exactly one way to arrange zero objects: do nothing.
Because of this, mathematicians define:
0! = 1
This convention appears throughout probability theory, statistics, algebra, and computer science.
Factorial Calculator — Values
| n | n! |
|---|---|
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
| 6 | 720 |
| 7 | 5,040 |
| 8 | 40,320 |
| 10 | 3,628,800 |
Factorials and Permutations
One of the most common applications of factorials is calculating permutations.
A permutation counts how many ways items can be arranged when order matters.
For example, if three people are competing for gold, silver, and bronze medals, the order matters significantly. Winning first place is different from winning second place.
The number of possible arrangements can be calculated using factorial formulas.
This is why factorials are heavily used in sports statistics, scheduling systems, ranking problems, and probability calculations.
Students working with permutations often use a Scientific Calculator alongside factorial functions when solving larger equations involving multiple operations.
Factorials and Combinations
While permutations care about order, combinations focus on selection.
Suppose you need to choose 3 students from a class of 10. The order of selection doesn't matter. Whether Alice is chosen before Bob or after Bob produces the same group.
Combination formulas rely heavily on factorials:
nCr = n! / [r!(n-r)!]
This formula appears in probability courses, data analysis, machine learning, and statistics.
Understanding factorials makes combination calculations much easier and helps students understand why these formulas work.
Frequently Asked Questions
What is a factorial in mathematics?
A factorial is the product of a whole number and every positive integer smaller than it. It is written using an exclamation mark (!). For example, 6! means 6 × 5 × 4 × 3 × 2 × 1 = 720. Factorials are widely used in probability, statistics, algebra, combinatorics, and computer science.
Can I calculate large factorials online?
Yes. This free online Factorial Calculator can calculate very large factorial values instantly without doing lengthy multiplication by hand. It also displays step-by-step working so you can understand how the result is obtained.
Why is 0! equal to 1?
Although it may seem unusual, mathematicians define 0! = 1. This rule keeps many important formulas in probability, permutations, combinations, and algebra mathematically consistent. It represents the single possible arrangement of zero objects.
Can factorials be calculated for negative numbers?
No. Standard factorials are only defined for non-negative integers (0, 1, 2, 3, ...). If you need factorial values for non-integer or negative numbers, mathematicians use the Gamma Function, which extends the factorial concept beyond whole numbers.
What is the largest factorial this calculator can handle?
The exact limit depends on your browser and device, but this calculator supports extremely large values that would be almost impossible to compute manually. Even when the answer contains hundreds or thousands of digits, the calculator processes it within seconds.
Where are factorials used in real life?
Factorials appear in many practical fields including statistics, probability, cryptography, artificial intelligence, scheduling, computer algorithms, genetics, operations research, and machine learning. They are especially important whenever different arrangements or selections need to be counted.
Related Calculators You May Find Useful
If you're solving probability or higher mathematics problems, several other tools can save time alongside this Factorial Calculator. For example, the Scientific Calculator is ideal for advanced equations involving exponents, roots, logarithms, and trigonometric functions.
When your calculations involve powers or repeated multiplication, try the Exponent Calculator. If you're working with very large numbers generated from factorials, the Scientific Notation Calculator converts lengthy values into an easier-to-read format.
Many combinatorics problems also require understanding precision. The Significant Figures Calculator helps round answers correctly, while the Log Calculator and Antilog Calculator simplify logarithmic calculations often used in advanced mathematics.
If your work includes number systems, the Number Base Converter, Binary Calculator, Hex Converter, and Roman Numeral Converter make conversions quick and accurate.
For arithmetic operations, you may also find the Long Division Calculator, Modulo Calculator, GCF Calculator, and Prime Number Checker useful when solving number theory or divisibility questions.
If you're studying statistics, the Average Calculator and Standard Deviation Calculator pair naturally with factorial calculations used in probability distributions and data analysis.
Final Thoughts
Whether you're preparing for an exam, solving a probability question, checking homework, or working on programming and data science projects, this free Factorial Calculator provides a fast, reliable, and beginner-friendly way to calculate n! values. Instead of performing long multiplication manually, simply enter your number to receive an accurate answer along with clear step-by-step calculations.
Because factorials grow incredibly fast, even relatively small numbers can produce enormous results. This calculator eliminates calculation errors, saves valuable time, and helps you understand the mathematics behind every answer. From basic classroom exercises to advanced combinatorics and statistics, it's designed to make factorial calculations simple, accurate, and accessible for everyone.
Reviewed by Kromo Tools Team
Building free, reliable online calculators and utilities for students, professionals, and everyday users. Last updated:
