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A factorial, denoted as n!, is a mathematical function that multiplies a whole number by every whole number below it down to 1. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. This concept is fundamental in fields like combinatorics, probability, and calculus for counting possible arrangements.
What is a Factorial Used For?
Factorials are rarely used in isolation but are the building blocks for calculating permutations and combinations. Permutations concern the number of ways to arrange items where the order matters, while combinations deal with groupings where the order does not matter. The formulas for both rely on factorials:
How Do You Calculate a Factorial by Hand?
Calculating a factorial involves a straightforward, sequential process. Let's break down how to calculate 9!.
What Are the Values of Common Factorials?
While small factorials are easy to compute, larger ones grow exponentially. A special case defined by mathematicians is 0! (zero factorial), which is equal to 1. The table below provides a quick reference for the first 15 factorials plus zero.
| n | n! (Formula) | Result |
|---|---|---|
| 0 | 1 (by definition) | 1 |
| 1 | 1 | 1 |
| 2 | 2 x 1 | 2 |
| 3 | 3 x 2 x 1 | 6 |
| 4 | 4 x 3 x 2 x 1 | 24 |
| 5 | 5 x 4 x 3 x 2 x 1 | 120 |
| 6 | 6 x 5 x 4 x 3 x 2 x 1 | 720 |
| 7 | 7 x 6 x 5 x 4 x 3 x 2 x 1 | 5,040 |
| 8 | 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 | 40,320 |
| 9 | 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 | 362,880 |
| 10 | 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 | 3,628,800 |
| 11 | 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 | 39,916,800 |
| 12 | 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 | 479,001,600 |
| 13 | 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 | 6,227,020,800 |
| 14 | 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 | 87,178,291,200 |
| 15 | 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 | 1,307,674,368,000 |
For values larger than 15, using a scientific calculator or software is essential due to the immense size of the results. Factorials for negative integers are undefined, but advanced mathematics uses the Gamma Function to extend the concept to decimal numbers.
In summary, understanding factorials is key to mastering more complex counting principles. Remember that the factorial of a number n is the product of all integers from 1 to n, and 0! is always 1. For practical application, use the table for quick reference and a calculator for larger computations.
