What is 36 factorial ?

The concept of a factorial in mathematics pertains to the product of a series of descending natural numbers from a particular number down to one. The factorial of 36, designated as 36!, involves the multiplication of all whole numbers from 36 to 1. This numeric operation is not only a test of the magnitude of numbers but also plays a substantial role in various mathematical and real-world applications due to its link to permutations and combinations.

The factorial of a number is calculated using the formula:

n! = n × (n-1) × … × 1

For the number 36, the factorial (36!) is computed by multiplying 36 by every number that precedes it down to 1. While the resulting product for 36! is extremely large, and impractical to calculate manually, it can be understood as a continuation of the factorial function’s fundamental concept.

Factorials, such as 36!, are instrumental in various fields including:

• Combinatorics: To calculate permutations, combinations, and configurations of sets.
• Probability Theory: To determine the probabilities of various outcomes in scenarios with numerous possible arrangements.
• Algorithms and Computer Science: Factorials can help optimize the computation time of certain processes that deal with combinations.
• Quantitative finance: In understanding results that have factorial components such as the Black-Scholes formula for options pricing.

These applications underscore the practical importance of understanding factorials in mathematical and real-world contexts.

Engage with these exercises to better grasp the concept of factorial:

1. Without calculating the entire product, determine whether 36! is an even or odd number.
2. How many zeros are at the end of the number 36! when it is written in full?
3. If you have 36 different books, in how many ways can you arrange them on a shelf?

Below are the solutions to the exercises:

1. 36! is an even number because it includes the multiplication of 2 (a factor of every even number).
2. To determine the number of zeros at the end of 36!, count the number of 5s present in the factorial sequence since each pair of 2 and 5 contributes to a trailing zero. In 36!, there are 7 zeros at the end.
3. Since every unique arrangement represents a permutation, there are 36! ways to arrange 36 different books on a shelf.