What is 200 factorial ?

A factorial is a mathematical operation denoted with an exclamation mark “!”, representing the product of all positive integers up to a certain number. The factorial of 200, or 200!, encompasses a vast multiplication that extends from 200 down to 1. Understanding 200! is significant in mathematics, particularly when we delve into high-level concepts such as combinatorics or statistical calculations involving large datasets.

The basic formula to calculate the factorial of a number is n! = n × (n-1) × … × 3 × 2 × 1. To apply this to 200, you would theoretically multiply all whole numbers from 200 down to 1. However, since the number is extremely large, this calculation isn’t typically done by hand but with the aid of computers or factorial calculators.

The factorial of 200 can be employed in several intriguing mathematical and practical applications. In combinatorics, it is used to determine the number of ways to arrange 200 different objects. In probability theory, factorials play a key role in computing permutations and combinations, which could be useful in advanced modeling or forecasting. Such large factorials are also common in algorithms and computational mathematics where an understanding of large scale data behavior is required.

• Compute the last two digits of 200!.
• Estimate the number of zeros at the end of 200!.
• Without calculating the exact number, explain if 200! is even or odd and why.

1. The last two digits of 200! are “00” because any factorial greater than 9! ends with at least two zeros.

2. The number of zeros at the end of 200! can be found by dividing 200 by 5 (and all multiples of 5) and summing up the quotients, due to the number of 5*2 pairs.

3. The number 200! is even because it includes the multiplication of many even numbers, including 2, which is the smallest even number.