Probability and Statistics > Basic Statistics> > What is a Factorial ! ?

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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 × 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 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:

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 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.

Check out our YouTube channel for short videos on stats topics.

My biggest challenge with chapter 2 was the factorials. The sixth-edition Elementary Statistics by Allan G. Bluman is well worth the $166.00 +. Factorials is defined and explained in chapter 4, section 5, pages 218 and 219. I have had to read and read this section several times and now I feel more comfortable with factorials. One of the ways to identify a factorial problem is with the factorial notation of the usage of the explanation point (s). A factorial is a series of positive integers that are used to calculate the outcome of how to measure an outcome, a good example is in the textbook, page 219: the number to the left of C and the number to the right of C determine how many factorials are needed to solve this equation and sometimes more than one equation is needed to solve a word problem. We use factorials without realizing it when we have an x amount of women and an x amount of men to be chosen to serve our country; and as to how many automobiles does car maker A, B or C market in a particular make, model and color. This is difficult to learn, but well worth the effort, sweat and tears!

The explanation provided helps but my biggest problem with factorials is figuring out my calculator. Possibly a calculator walkthru on how to run factorials would be helpful. Maybe something like, if you have this type of calculator, this is what you need to push in order to find the factorial key

Angie,

Good point. I’m planning to put together something for the different calculators at a later date.

Have you tried the permutations and combinations calculator?

Stephanie

I also could definitely use some help in figuring out the calculator. There are functions on there that I have never had to use before and am very unfamiliar. I think factorials were better explained here in the blog than in the book we were required to buy-it unfortunately was of little help.

I too am having a lot of problems with factorials. In this section is suggest that we use Google and it will be able to help find the answer, but I am still lost. For example, I entered 22! / (22-4)!4! into google and got a bunch of sites with pharmacy codes and nothing pertaining to help on this question. Do I need to get a scientific calculator?

I think you may be entering the formula into Google incorrectly. Check out the formula in the article–it’s different from the 22! / (22-4)!4! you entered into Google. With any calculator (including scientific ones), you would need to use parentheses: 22! / ((22-4)! * 4!). Also note that I added a multiplication sign * because Google doesn’t understand that 4!4! means 4! * 4!.

Hope that helps! You are certainly welcome to buy a calculator if you feel that it will help you, but you don’t have to have one for this course.

I also had issues trying to figure this stuff out on my calculator. I think I sat there for hours trying to figure out how to convert by decimal answers into fractions HAHA But I figured it out!

I just used google using your steps and it worked GREAT!!! I now have a better understanding of what I am doing for this problem. Thank you!

Awesome…glad it helped!

Stephanie

Great. I am glad the site is helping a little. Stats can be a difficult subject to tackle!

Stephanie

I was manually entering the counting problems into my calculator one at a time (22x21x20 etc.) which worked but it took forever! I wish I would have read this post before Chapter was over. Google is a great tool. I am going to write down the calculations in my notes for 1!-30! just for an easy reference. Thanks everyone!

I wish I would have seen this earlier. I had to google what the ! meant. Then I didn’t realize until today my calculator had this feature. Maybe it will get easier now.

Glad it helped :)

I just used Google to do this and it worked and I liked how it showed to input it in the calculator because I will not always have access to a computer when doing these sorts of problems. One suggestion for your explanation up top would be to explain the steps to put it in the calculator. For someone like me, who is statistics illiterate, I need every single tiny step laid out for me or I will not understand it. Thank you for directing me to this section to help me with the factorial problems.

google really did help with this! it made it much easier.

Wish I would have read this a couple of weeks ago, maybe this will help me now, I will work on this tomorrow. This class is pretty difficult so far. Thanks goodness I am only taking 2 classes, because I could have never handled any more than that.

Easy & good