What is 54 factorial ?

The concept of factorial in mathematics refers to the product of all positive integers up to a given number, denoted as n!. This operation is crucial for various mathematical fields, such as combinatorics and probability theory. Specifically, 54 factorial, written as 54!, signifies the product of all integers from 1 to 54. Understanding the factorial of 54, due to its enormity, illuminates the scale of combinations and permutations possible in large sets.

The formula to calculate the factorial of a number is n! = n × (n-1) × … × 1. To calculate 54!, you would multiply 54 by every number less than it down to 1. While the direct multiplication for such a large number is impractical to display in full, the process would look something like this:

54! = 54 × 53 × 52 × … × 3 × 2 × 1

Calculating 54! is typically done with the aid of computational tools due to the extensive number of multiplications involved.

The factorial function primarily applies to fields where arrangement and selection are essential. The factorial of 54 can be used in:

• Calculating permutations and combinations in large data sets and samples.
• Assessing probabilities in complex situations, such as lottery games involving 54 unique numbers.
• Queue theory and other parts of discrete mathematics.
• Cryptographic algorithms that rely on large factorial numbers for encryption keys.

Consider the following exercises related to the factorial of 54:

1. Calculate the number of zeroes at the end of the 54!.
2. If a group contains 54 unique items, in how many ways can you pick 10 without replacement?

Here are the solutions to the above exercises:

1. The number of zeroes at the end of 54! is determined by the number of 10’s we can factor from the factorial, which is primarily based on how many times the factors 5 and 2 are present. After calculation, we find there are 13 zeroes at the end of 54!.
2. To pick 10 items from 54 without replacement, you can use the formula for combinations, given as C(n, r) = n! / [r!(n – r)!]. Here we have C(54, 10) which evaluates to a sizable number symbolizing the vast array of possibilities.