One of the fundamental concepts in mathematics taught in schools is the factorial. It involves multiplying all numbers from 1 up to a specified number and then adding them together. For instance, the factorial of 4 is calculated as 1 * 2 * 3 * 4, resulting in 24, while the factorial of 7 equals 1 * 2 * 3 * 4 * 5 * 6 * 7, totaling 5,040. Understanding factorials becomes particularly useful when examining larger values, such as 100!.
Utilizing Calculators
Calculators prove invaluable when dealing with complex mathematical problems involving numerous variables. Employing a calculator to compute 0.01 x 1 x 2 x 3 . . . 97 x 98 significantly saves time and mental effort. However, not all calculators handle factorials equally well, especially when dealing with exceedingly large numbers like 100!.
Applications of Factorial
The factorial function finds applications in diverse mathematical areas. It serves as the foundation for calculating permutations, representing the different ways of arranging distinct objects in a sequence. Factorials are extensively used in combinatorics formulas for various object orderings. For instance, binomial coefficients (n choose k) compute combinations of k elements from a set of n elements and are derived from factorials. Additionally, factorials are integral to Stirling numbers of the first kind and counting permutations of n with the same number of cycles. Another combinatorial application involves computing derangements, which are permutations that leave no element in its original position. The number of derangements of n items is approximately n! / e.
Factorials also play a crucial role in the binomial theorem, which expands the powers of a sum using binomial coefficients. They feature in coefficients linking specific families of polynomials, such as Newton’s identities for symmetric polynomials. Moreover, factorials correspond to the orders of finite symmetric groups, contributing to their application in algebraic permutation computations. In calculus, they appear in Fadi Bruno’s formula for iterated derivatives and frequently emerge in power series denominators, especially those related to exponential functions in mathematical analysis.
Calculation Process
To compute the factorial of a positive integer n (n!), one multiplies all positive integers less than or equal to n. For example, 5! equals 1 × 2 × 3 × 4 × 5, resulting in 120. The factorial symbol varies across different algebraic systems, with the exclamation mark (!) commonly used. Unicode adopted the exclamation mark as the factorial sign in Unicode 2.1 (1999), represented as !!! ≡ 1·2·3·4·5 = 120.
Understanding 1 Factorial
Determining the factorial of 1 may seem straightforward, but it’s a question that often raises uncertainties. The factorial definition states that n! equals the product of all integers from 1 to n. Thus, 1 factorial equals 1.
Table of factorials up to 30
| n | n! |
| 1 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
| 6 | 720 |
| 7 | 5040 |
| 8 | 40320 |
| 9 | 362880 |
| 10 | 3628800 |
| 11 | 39916800 |
| 12 | 479001600 |
| 13 | 6227020800 |
| 14 | 87178291200 |
| 15 | 1307674368000 |
| 16 | 20922789888000 |
| 17 | 355687428096000 |
| 18 | 6402373705728000 |
| 19 | 121645100408832000 |
| 20 | 2432902008176640000 |
| 21 | 51090942171709440000 |
| 22 | 1124000727777607680000 |
| 23 | 25852016738884976640000 |
| 24 | 620448401733239439360000 |
| 25 | 15511210043330985984000000 |
| 26 | 403291461126605635584000000 |
| 27 | 10888869450418352160768000000 |
| 28 | 304888344611713860501504000000 |
| 29 | 8841761993739701954543616000000 |
| 30 | 265252859812191058636308480000000 |
Result and Conclusion
Upon calculating 100!, the result is a staggering number expressed in scientific notation: 2.113857 x 10^157. For computations involving factorials beyond double digits, one can multiply each digit by every other digit in the number and then sum the products. For instance, to calculate 14!, multiply 1 by 4 to get 4, then multiply 4 by 1 to again get 4, and finally, add these digits together to yield 42.
