Factorial Calculator N and R

Factorials are fundamental in combinatorics and probability. This factorial calculator n and r helps you compute factorials for any positive integer, and explains how to use them in permutations and combinations.

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. Factorials are widely used in combinatorics, algebra, and probability theory.

Factorial definition

n! = n × (n-1) × (n-2) × ... × 1

For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials grow very rapidly with increasing n, which is why they're important in probability calculations where large numbers of possible outcomes exist.

Factorial formula

The factorial formula is simple but powerful. For any positive integer n:

Factorial formula

n! = n × (n-1)!

With the base case: 0! = 1

This recursive definition makes it easy to compute factorials using programming or repeated multiplication. The factorial function is strictly increasing for n ≥ 1, meaning each factorial is larger than the previous one.

Permutations and combinations

Factorials are essential for calculating permutations and combinations:

Permutations formula

P(n,r) = n! / (n-r)!

Combinations formula

C(n,r) = n! / (r! × (n-r)!)

These formulas show how factorials help count arrangements and selections without regard to order. The factorial calculator n and r can help you compute these values quickly.

How to calculate factorial

Calculating factorials manually is straightforward for small numbers but becomes impractical for large n due to the rapid growth of the result. Here's how to compute factorials:

Start with the given number n
Multiply n by each integer from n-1 down to 1
Continue multiplying until you reach 1
The final product is n!

Note

Factorials of numbers greater than 20 quickly exceed the storage capacity of standard data types in programming languages, so most factorial calculators limit n to 20 or less.

Factorial examples

Here are some factorial calculations:

n	n!
0	1
1	1
2	2
3	6
4	24
5	120
6	720

These examples show how quickly factorials grow. For instance, 6! = 720, which is already a large number, while 10! = 3,628,800.

Factorial applications

Factorials have many practical applications in mathematics and computer science:

Combinatorics: Counting permutations and combinations
Probability: Calculating probabilities in discrete distributions
Algebra: Expanding polynomials and solving equations
Computer science: Algorithms and data structures
Physics: Quantum mechanics and statistical mechanics

Understanding factorials is essential for anyone working in these fields, as they provide a foundation for more advanced mathematical concepts.

FAQ

What is the factorial of 0?: The factorial of 0 is defined as 1. This base case is necessary for the recursive definition of factorials to work properly.
How do I calculate large factorials?: For large factorials, it's often better to use a programming language or specialized mathematical software that can handle very large numbers.
What's the difference between permutations and combinations?: Permutations consider the order of items, while combinations do not. The formulas differ by whether you divide by r! in the denominator.
Can factorials be negative?: No, factorials are only defined for non-negative integers. Attempting to calculate the factorial of a negative number will result in an error.
Where are factorials used in real life?: Factorials are used in probability calculations, combinatorial problems, and various scientific and engineering applications.