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## What is a Factorial ! ?

**Factorials **are introduced in **algebra**. They are also used in stats classes, especially in permutations and combinations calculations.

When you see the ! symbol after a number, that means it’s a factorial:

- 6! is “six factorial.”
- 3! is “three factorial.”

The “Mathy” definition: Factorials are products of every whole number from 1 to n. For example, if n is 3 then 3! is 3 x 2 x 1 = 6. It’s really just a shorthand way of writing numbers. For example, instead of writing 479001600, you could write 12! instead (which is 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1).

### Using the TI 83

Most calculators have a button for this. It’s usually hidden in a menu somewhere. On the TI 83, you can find it in the “PRB” menu. Here’s how to find six factorial (6!) on the TI-83:

- On the home screen, press [CLEAR]
- Press [6] [MATH] and then press the left arrow key to select the PRB menu.
- Press [4] for !.
- Press [ENTER] to display the answer of 720.

### Using Google

If you’re on the internet (which you probably are if you’re reading this), Google can also do the work for you.

- Go to the search bar at Google.com
- Type in a factorial, such as 12!
- Press enter
- 12 ! = 479 001 600

Google can also figure out more complicated factorials for you, like 36! / (12-10)!6!. Make sure you put in parentheses and a multiplication sign (just as you would on any basic calculator). Like this:

36! / ((12-10)! * 6!) = 2.58328699 × 10^{38} (☨)

**Google Calculator Tip:** To multiply using Google, use an asterisk (*) instead of a “×” symbol.

### What is a factorial used for in stats?

In algebra, you probably encountered ugly-looking factorials like (x – 10!)/(x + 9!). Don’t worry; You won’t be seeing any of these in your beginning stats class. Phew! The *only* time you’ll see them is for permutation and combination problems.

The equations look like this:

And that’s something you can input into your calculator (or Google!).

## 10 Factorial

10 factorial is equal to 3,628,800.

**Why?**

10 factorial is just another way to write 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1.

In other words, 10 factorial is just multiplying 10 by a descending series of natural numbers. You’ll usually see it written as 10!. The exclamation point just means factorial in the same way that π means pi or *e *means Euler’s number.

## 10 Factorial Seconds

Interestingly, 10 factorial is exactly the number of seconds in 6 weeks.

60 seconds in a minute

x 60 seconds in an hour = 3600

x 24 hours in a day = 86400

x 7 days in a week = 604800

x 6 weeks = 3,628,800.

A reverse way to do this is to factor 3,628,800: you’ll get all the numbers from 1 to 9.

It’s one of the only factorials that represents something in real life that you can relate to* (according to John D Cook, Avogadro’s constant—an important number in chemistry—comes close at around 24! but it isn’t exact).

*There are a few other factorials that relate to time:

- 4! seconds = 24 seconds.
- 5! seconds = 2 minutes.
- 6! seconds = 12 minutes.

## How to Find 10 Factorial

You have several ways to find 10 factorial:

- Multiply 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 on any calculator.
- Type “10 Factorial” into the Google search box. Google will give you the answer.
- Find the button on your calculator (the ! button). You may have to search in the catalog. On the TI 83, you can find the ! button by pressing MATH and then the left arrow key. Scroll down to !. Use the ! key after a number on the home screen. For example, type 10! and then press Enter.
- Use an online calculator, like this one.Just type 10 into the box and then press Calculate.
- The long way, using unit analysis and prime factorization. The image below shows the process. Thanks to Yozh.org for this procedure.

## Factorials 1! – 30!

01! = 1

02! = 2

03! = 6

04! = 24

05! = 120

06! = 720

07! = 5040

08! = 40320

09! = 362880

10! = 3628800

11! = 39916800

12! = 479001600

13! = 6227020800

14! = 87178291200

15! = 1307674368000

16! = 20922789888000

17! = 355687428096000

18! = 6402373705728000

19! = 121645100408832000

20! = 2432902008176640000

21! = 51090942171709440000

22! = 1124000727777607680000

23! = 25852016738884976640000

24! = 620448401733239439360000

25! = 15511210043330985984000000

26! = 403291461126605635584000000

27! = 10888869450418352160768000000

28! = 304888344611713860501504000000

29! = 8841761993739701954543616000000

30! = 265252859812191058636308480000000

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