# What is a Factorial ! ? …and How to Solve Them

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## What is a Factorial ! ?

Factorials are introduced in algebra. They are also used in stats classes, especially in combination and permutation calculations.

When you see the ! symbol after a number, that means it’s a factorial:
6! means “six factorial.”
3! means “3 factorial.”
It’s a shorthand way of writing numbers. For example, instead of writing 479001600, you could write 12! instead.

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.

### What is a Factorial? Using the TI 83

Most calculators have a factorial button. 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.

### What is a Factorial? Using Google

If you’re on the internet, Google can also do the work for you!

• Go to the search bar at Google.com
• Type in a factorial, such as 36!
• 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 × 1038 (☨)

### 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 statis class. Phew! The only time you’ll see them is for permutation and combination problems.
How to Solve Permutations and Combinations.
The equations look like this:

## 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.
• Find the factorial 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 factorial 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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