What is 26 factorial ?

In mathematics, a factorial is the product of all positive integers up to a certain number, symbolically represented by an exclamation point following the number. The factorial of 26, denoted as 26!, involves multiplying 26 by all the positive integers less than it down to 1. Understanding the factorial of 26 is significant in mathematics as it relates to a variety of fields such as combinatorics, where it can denote the number of possible permutations of a 26-element set. This has practical applications in areas like cryptography, where the strength of a 26-letter code could be assessed by considering the number of possible combinations, which is exactly 26!.

To calculate the factorial of a number such as 26, we utilize the basic mathematical formula: n! = n × (n-1) × … × 3 × 2 × 1. Applying this formula to 26, the calculation would start by multiplying 26 with every number less than 26 down to 1. Although the resulting number is quite large, the process remains straightforward. For a comprehensive example:

• 26! = 26 × 25 × 24 × … × 3 × 2 × 1
• Begin with the largest number, 26, and multiply it by the next smallest, 25, and continue this process until reaching 1.

The factorial function, particularly the factorial of 26, has intriguing applications across various domains. In combinatorics, 26! is the number of possible orders for 26 distinct items. In probability theory, factorials determine combinations and permutations, which help in calculating the likelihood of various outcomes. For instance, the sequencing of the 26 letters in the English alphabet can be calculated using factorials, which is particularly relevant in the field of cryptography for creating complex encryption codes.

Test your understanding of factorials with these exercises:

• Exercise 1: Calculate the last two digits of 26!.
• Exercise 2: If there are 26 teams in a league, in how many ways can a first, second, and third place be awarded?
• Exercise 3: Assuming a 26-character password, using each character only once, how many possible passwords can you make?

Here are the solutions to the exercises:

• Solution to Exercise 1: The last two digits of 26! are “00” because the factorial includes the term 10, which introduces a trailing zero, and any subsequent multiplication by an even number will add another zero.
• Solution to Exercise 2: The number of ways three teams can win first, second, and third places from 26 teams can be calculated as 26P3 = 26! / (26-3)! = 15,600 ways.
• Solution to Exercise 3: The number of possible 26-character passwords is 26!, because each character can only be used once, leading to a permutation problem.