What is the Factorial of 100?

The factorial of 100, written as 100!, is a mathematical expression that means multiplying every positive whole number starting from 100 down to 1. In simpler terms, 100 factorial is:

100! = 100 × 99 × 98 × 97 × ... × 3 × 2 × 1.

This operation is called a factorial and is important in various areas of mathematics like probability, combinations, and algebra. The factorial function grows very quickly, so the result of 100! becomes an extremely large number that is difficult to write or calculate manually.

To understand what "100 factorial" means, let's break it down step by step:

• Factorial of any number n (denoted as n!) is the product of all positive integers up to n.
• For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
• For 100!, you multiply all numbers from 100 to 1.

Since 100 is a large number, 100! becomes extraordinarily big. The exact value of 100! is:

93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

This number has 158 digits! Writing or calculating it without a calculator or computer is nearly impossible.

To make it easier to express such large numbers, mathematicians also use scientific notation. The factorial of 100 in scientific notation is approximately:

100! ≈ 9.332621544 × 10¹⁵⁷

This means the number is 9.3326 followed by 157 zeros.

Another interesting fact about 100! is that it ends with 24 trailing zeros. Trailing zeros occur because of factors of 10 in the multiplication, which come from pairs of 2 and 5. Since there are more 2s than 5s, the number of trailing zeros depends on the count of 5s in the multiplication series.

Why is factorial important? Factorials are widely used in areas such as:

• Combinatorics: Counting how many ways you can arrange or choose items (permutations and combinations).
• Probability: Calculating probabilities in complex scenarios.
• Algebra and calculus: Expansion of functions like exponential series.
• Computer science: Algorithms and complexity analysis.

In summary, "What is 100 factorial?" is asking about the product of all positive integers from 1 to 100. The answer is a very large number with 158 digits, represented in short form as about 9.33 × 10¹⁵⁷. Due to its size, factorials like 100! are typically handled using computers or scientific notation, not manual calculation.