# What is the factorial of 100?

The factorial of 100 is a very large number, approximately equal to 9.33262154 times 10 raised to the power of 157. It's written as 100! (100 factorial). Due to its immense size, it's impractical to write out the entire number.

What is a factorial?

A factorial, denoted by n!, is the product of all positive integers less than or equal to a non-negative integer n. In simpler terms, it's the result of multiplying a number by all the whole numbers below it down to 1.

How do you calculate a factorial?

For example, 5! (factorial of 5) is calculated as 5 x 4 x 3 x 2 x 1 = 120. There are calculators and computer programs that can handle very large factorials, but for smaller values, manual multiplication works well.

What is the factorial of 0?

By convention, 0! (factorial of zero) is defined as 1. This is because 0! represents the number of ways to arrange zero objects, and there's only one way to arrange nothing - by having nothing there!

What are factorials used for?

Factorials have applications in various fields like combinatorics (counting arrangements and combinations), probability theory, and even cryptography. They help calculate things like the number of possible hands in a card game or the number of unique permutations of items.

Do factorials ever become negative?

No, factorials are only defined for non-negative integers. Since you're multiplying positive numbers, the result will always be positive (or zero in the case of 0!).

Is there a limit to how big a factorial can get?

Factorials grow very rapidly as the number increases. Even with calculators and computers, calculating factorials of very large numbers becomes impractical due to the immense size of the result.

Is there a shortcut for calculating factorials?

There's no general shortcut for calculating factorials by hand except for memorizing the values of small factorials (like 1! to 5!). However, mathematicians have devised advanced techniques for estimating factorials of large numbers.

What's the difference between factorial and permutation?

Factorial simply tells you the total number of arrangements possible for a certain number of objects. Permutation considers the order of arrangement, so there are n! permutations of n distinct objects.

What's the relationship between factorials and recursion?

Factorial can be defined recursively, meaning you define n! based on the value of (n-1)!. This is a common approach in computer science for calculating factorials.

Are factorials related to other mathematical concepts?

Yes! Factorials are linked to concepts like gamma function, which extends the idea of factorial to non-integer values. They also play a role in Stirling numbers, used for counting partitions of sets.