What is 500 factorial ?

In mathematics, a factorial is the product of all positive integers less than or equal to a given number. The factorial of 500, denoted as 500!, is a vast number that extends beyond typical calculating capabilities. Factorials, especially those of large numbers like 500, are significant because they play a crucial role in various mathematical fields including combinatorics, probability, and series expansions.

To calculate the factorial of a number, you multiply the number by every positive integer below it down to 1. The basic formula is:

n! = n × (n-1) × … × 3 × 2 × 1

Applying this to 500, we multiply all numbers from 500 down to 1. Since this process would result in an extremely large number, 500 factorial is typically not calculated explicitly but represented symbolically or computed using specialized software.

Factorials are used to determine the number of ways in which a set of objects can be arranged (permutations). The factorial function, particularly for a large value like 500, is used in algorithms that solve combinatorial problems, in statistical formulas to define probabilities, and is of theoretical interest in understanding the behavior of large numbers in mathematics.

Here are some exercises to help you explore the concept of factorial:

• Even without calculating the exact number, explain why the factorial of 500 ends with at least one zero.
• What is the number of possible combinations if you have a deck of 500 different cards?

The factorial of 500 ends with zeros because the number contains multiple factors of 10, which is the product of 5 and 2, both abundant in the prime factorization of 500!.

For a deck of 500 different cards, the number of possible combinations is 500! This concept is essential to understand the true randomness and complexity in shuffling a card deck.