Factorial
Definition
The product of all positive integers less than or equal to a given number n, written as n! (e.g., 5! = 120).
Why is Factorial Important?
Factorial is a foundational mathematical concept used across science, engineering, finance, and everyday problem-solving. From analyzing data sets to optimizing business decisions, this concept provides the analytical framework needed to interpret quantitative information accurately.
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What is a Factorial?
The factorial of a non-negative integer n, written as n!, is the product of all positive integers from 1 to n. Factorials grow extraordinarily fast and are foundational in counting, probability, permutations, and combinations.
Factorial Values
| n | n! | Value |
|---|---|---|
| 0 | 0! | 1 (by definition) |
| 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 |
| 10 | 10! | 3,628,800 |
| 12 | 12! | 479,001,600 |
| 15 | 15! | 1,307,674,368,000 |
| 20 | 20! | 2,432,902,008,176,640,000 |
Where Factorials Are Used
- Permutations: n! = number of ways to arrange n distinct objects (10 books on a shelf: 10! = 3,628,800 arrangements)
- Combinations: C(n,r) = n! / (r!(n-r)!) โ choosing r items from n without regard to order
- Probability: Many probability distributions use factorials (binomial, Poisson)
- Taylor series: sin(x), cos(x), eหฃ are defined using factorial denominators