Summary:

- Factorials are just a shorthand way to multiply lots of numbers together!

- You just keep multiplying from the number you're given down to "1".

- No magic is involved!

- Factorials are just a shorthand way to multiply lots of numbers together!

- You just keep multiplying from the number you're given down to "1".

- No magic is involved!

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## The Math Tutor - Austin

Factorials!

Factorials are just shorthand!

At some point in math, you're going to encounter formulas where you're supposed to multiply a lot of consecutive numbers together.

For example, you might want to multiply all of the numbers from 10 down to 1 together, like this:

10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3628800

Easy concept, right? Except for two things:

- It takes too long to write it down as "10 x 9 x 8 ...", etc.

- Just writing the answer takes a long time to write, too!

Wouldn't it be nice to have a shorthand way of writing down all of that multiplication?

Well, turns out that Mathematicians already invented a symbol for that. The product of all of the numbers from "10" down to "1" is referred to as "10 factorial", and Mathematicians write it symbolically like this:

10!

Just be careful to read that as "10 factorial" - if you think, instead, that it just means that I'm excited about the number 10, you missed the point ;)

Here are some examples of how to decode the notation:

5! = 5 * 4 * 3 * 2 * 1

50! = 50*49*48*47 * ... * 4 * 3 * 2 * 1

3! = 3 * 2 * 1

2! = 2 * 1

1! = 1

Easy, right?

A trick - what if I don't want to multiply all the way down to 1??

Ah - good question! What if you just wanted to multiply some of the consecutive numbers together - like, say, from 29 down to 12?

i.e.

29 * 27 * 26 * 25 * 24 * 23 * 22 * 21 * 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12

That takes up a lot of room too, right? And - it isn't exactly a factorial, is it? I mean, 29! would have a lot more numbers:

29! = 29 * 27 * 26 * 25 * 24 * 23 * 22 * 21 * 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * ... * 3 * 2 * 1

Do we have to invent another symbol for this idea as well?

The answer is - not really... Instead, we could use two factorials together in a fraction, so that we can cancel out the extra numbers we don't want...

i.e.

29!/11! = 29 * 27 * 26 * 25 * 24 * 23 * 22 * 21 * 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12

Because dividing by 11! canceled out all of the numbers from 11 on down to 1.

This is a good trick to remember if you want to save yourself some ink and paper!!