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Algebra II Textbook26 chapters | 256 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn about factorial notation. Understand what it means so that you can handle it like a pro. Also see what happens when we divide factorials.

What does the exclamation mark mean to you? We usually use it to make a statement such as 'Hey man, I got this!' Well, guess what? We also have a use for this mark in math. We call it 'factorial notation.' The **factorial notation** is the exclamation mark, and you will see it directly following a number. For example, you will see it as *5!* or *3!*. You read these as 'five factorial' and 'three factorial.' You will see your factorial notation when doing problems that ask you about the number of possible combinations. For example, factorial notation is used to find how many different ways you can arrange a blue marble, a red marble, and a green marble.

What does this factorial notation mean? **Factorial** means that we multiply all the integers less than or equal to our chosen number. So, 5! means that we multiply five times four times three times two times one, *5*4*3*2*1*, all the numbers less than or equal to our chosen number. Do you see how we just started with our 5 and kept multiplying it with the numbers we use to count down all the way to 1? This is what factorial is about. So, 3! is three times two times one, *3*2*1*. We just multiply it out and we have what our factorial equals. So, 5! equals 120, and 3! equals 6.

So, you can think of the factorial as telling you that 'you've got this!,' that you've got all the numbers less than or equal to your chosen number!

Now, what happens when we divide factorials? Like, what happens when we divide five factorial by three factorial, *5!/3!*? Well, let's see. First, we write out what each factorial means. So, we have *5*4*3*2*1/3*2*1*. Well, now we see that there are some things we can cancel out since we have the same number in the numerator AND the denominator. We can cancel out the threes and the twos and the ones. So, we are left with *5*4*. Well, isn't that interesting? Our problem got a lot simpler. We know that 5*4 equals *20*, and we are done.

So, what happens when our denominator is larger than our numerator? What if we divided three factorial by five factorial? Well, we would have *3!/5!*. Written out we have *3*2*1/5*4*3*2*1*. Canceling out the numbers that are common to both the numerator and denominator, we have *1/5*4*. I left the one in the numerator because we know from algebra that if everything cancels out, there is always a 1 there, as in x/x = 1. Now, to finish evaluating our factorial, we multiply out the 5*4 to get 20. We keep our answer in fraction form. So, our answer is *1/20*.

Let's look at some more examples.

We now know what 5! and what 3! equal, so what about *4!*? Well, we would do the same as we did with the other factorials. We write it out to get *4*3*2*1*. Then we multiply all the numbers out to get *24*. So, 24 is our answer.

How about *2!*? It is *2*1* which equals *2*.

*1!* is just *1*.

Now what about *0!*? This is a unique case! We're not going to get into the whole why it is, just know that 0! is *1*. We will keep it simple.

So, what have we learned now? We've learned that the **factorial notation** is the *exclamation mark*, and we see it directly following a number. For example, we see it as *5!* or *3!*. We read these as 'five factorial' and 'three factorial.' **Factorial** means that we *multiply all the integers less than or equal to our chosen number*. When we divide two factorials, we write out both factorials and we see what numbers we can cancel. Then after we have cancelled out all the numbers that we can, we multiply the numbers that are left, and if we have numbers left in the denominator, we leave our answer in fraction form. And 0! is always 1.

After you've completed this lesson, you'll have the ability to:

- Define factorial and identify the math symbol that indicates factorial
- Explain how to multiply and divide two factorials
- Recall what 0! is

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Algebra II Textbook26 chapters | 256 lessons

- Introduction to Sequences: Finite and Infinite 4:57
- How to Use Factorial Notation: Process and Examples 4:40
- Arithmetic Sequences: Definition & Finding the Common Difference 5:55
- How and Why to Use the General Term of an Arithmetic Sequence 5:01
- The Sum of the First n Terms of an Arithmetic Sequence 6:00
- Understanding Arithmetic Series in Algebra 6:17
- Working with Geometric Sequences 5:26
- How and Why to Use the General Term of a Geometric Sequence 5:14
- The Sum of the First n Terms of a Geometric Sequence 4:57
- Understand the Formula for Infinite Geometric Series 4:41
- Using Recursive Rules for Arithmetic, Algebraic & Geometric Sequences 5:52
- Using Sigma Notation for the Sum of a Series 4:44
- Mathematical Induction: Uses & Proofs 7:48
- How to Find the Value of an Annuity 4:49
- How to Use the Binomial Theorem to Expand a Binomial 8:43
- Special Sequences and How They Are Generated 5:21
- Go to Algebra II: Sequences and Series

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