What is 20 factorial ?

In mathematics, a factorial is a function that multiplies a given number, n, by every whole number less than n, down to 1. The factorial of 20, denoted as 20!, holds significance for its use in various mathematical fields and its ability to represent an extraordinarily large number. Understanding the factorial of 20 can help in solving complex combinatorial problems and in analyzing data sets in statistics.

The formula to determine a factorial is quite straightforward: n! = n × (n-1) × … × 1. For the number 20, you would calculate the factorial by starting with 20 and multiplying it by every number less than it down to 1.

For example:

• 20! = 20 × 19 × 18 × … × 1

This results in a very large number: 2,432,902,008,176,640,000.

Factorial functions, such as the factorial of 20, are utilized in a variety of mathematical areas, including:

• Combinatorics: To determine the number of ways to arrange or choose items.
• Probability Theory: In calculating probabilities where order is important.
• Series and Mathematical Constants: In the expansion series to approximate constants such as e or pi.

For instance, if you were to shuffle a deck of 20 unique cards, the total number of possible arrangements would be exactly 20!.

Challenge yourself with these exercises to deepen your understanding of factorials:

• 1. If one were to list all the possible permutations of 20 unique items, how many permutations would fall under number 20!?
• 2. What is the sum of the digits of 20!?

Here are the solutions to the exercises provided:

• 1. There are exactly 20! permutations of 20 unique items, which is 2,432,902,008,176,640,000.
• 2. The sum of the digits of 20! cannot be easily computed manually due to its enormous magnitude and would typically require a computer program to calculate.