# Factorials? What are They? A Simple Kind of Mathematics Shorthand

What are factorials? A variable is a symbol, often written as a letter of the alphabet that stands for a number that can vary in value. For example, take your age. That varies every year, doesn’t it? This year your age may be, say 21. If so, in 365 days your age will be 22. Another 365 days after that and your age will be 23.

Thus age is a function of time. For you, we can write right now:

A = 21

If the number of years that pass equals n, then for next year, n = 1 and

An = 21 + n

So,

A₀ = 21 + 0 = 21
A₁ = 21 + 1 = 22
A₂ = 21 + 2 = 23
A₃ = 21 + 3 = 24

and so on.

## What Are Factorials?

Now there is nothing special about the letter A. It can be replaced by any other symbol the reader may choose. So let’s choose a new variable symbol, F. We choose to follow this symbol with, of all things, an exclamation point (!). We define this symbol to mean the mathematics operation factorial. What is a factorial?

Written F! it means a product of the form

F(F‒1)(F‒2)(F‒3)… until we reach the value 1.Thus, if F = 7,

F! = 7(7‒1)(7‒2)(7‒3)(7‒4)(7‒5)(7‒6) = 7(6)(5)(4)(3)(2)(1) = 5040

Problem: Solve these factorials: F = 3, F = 5, F = 11. Write out the steps.