What is 100 factorial ?

In mathematics, the factorial of a non-negative integer is the product of all positive integers less than or equal to that particular number. When we talk about the factorial of 100, it’s significant because it represents an incredibly large value, and it’s essential for understanding concepts in combinatorics, probabilities, and even some functions in calculus. The factorial of 100, denoted as 100!, is particularly noteworthy in statistics and pure mathematics, revealing the vast number of ways to organize or select items from a large set.

To calculate the factorial of 100, we use the basic formula: n! = n × (n-1) × … × 1. If we were to apply this to 100, the calculation would be a large undertaking, as 100! equals 100 multiplied by every number less than 100 down to 1. For practical purposes and to prevent overwhelming calculations, we typically use computational tools or software for accurate computation of such large factorials.

The factorial of 100 can be used in several areas including combinatorics, where it helps to determine the number of ways to choose a subset of items from a larger set, or in arrangements where the order is significant. It’s also crucial in probability theory to calculate the likelihood of complex events. Additionally, in the realm of algorithm design, understanding the computation of factorials is fundamental when analyzing the efficiency and performance of recursive algorithms.

• Exercise 1: Estimate the number of trailing zeros in 100! without calculating the whole factorial.
• Exercise 2: If a classroom has 100 students, in how many ways can a president, vice president, and secretary be chosen?

Solution to Exercise 1: The number of trailing zeros in 100! is determined by the number of times 10 is a factor in the product, which in turn is determined by the instances of 2 and 5 pairings. As there are more 2s than 5s, we count the number of 5s, which is 24.

Solution to Exercise 2: Choosing 3 people out of 100 for the positions of president, vice president, and secretary can be calculated as a permutation: 100P3 = 100! / (100-3)! = 970,200.