Hey, everyone, as you work through different problems, you may come across some that have a number or a variable or an expression followed by an exclamation point. And it's not just that we're really excited to write that number because this exclamation point actually represents a specific type of operation called a factorial. Now, this might seem a bit strange at first because you're not used to using punctuation as a math operation. But here I'm going to walk you through exactly what a factorial is and how to simplify different factorial expressions. And soon this will be just like any other operation like addition or subtraction. So let's go ahead and get started. Now, I said that a factorial was a specific type of operation and the operation that it represents is multiplying all whole numbers from some specific number down to one. So for example, if I took four factorial I would start at four and then count down to one, multiplying all of those whole numbers along the way. So counting down to one, I get 43 to 1 and multiplying all of those numbers together would give me the factorial of four. Now, fact is always going to be represented with an exclamation point, whether it be in sequences, series combinatorics or probability. So now that we know what a factorial is, let's go ahead and calculate some. Now, I'm gonna skip zero factorial for now because it's a little bit strange, but we're going to come back to it. So let's go ahead and start with one factorial. Now, for one factorial, I can't really count down anywhere. So one factorial is simply going to be equal to one. Now, looking at two factorial, I can count down here. So I'm gonna go two times one and two factorial is then going to be equal to that, which is two. Now for three factorial, I'm going to count down from three multiplying three times two times one, which will give me a value of six. So three factorial is equal to six. Now, looking here at four factorial, I'm going to start at four, which we already saw this and multiply times three times two times one. Now multiplying all of this out will give me a value of 24. Then looking at five factorial, now I'm left to multiply five times four times three times two times one to give me a value of 120. Now, something that you may have noticed here is all we're really doing is taking some new number and then multiplying it by the previous factorial that we saw and this happens for every factorial. So each factorial is simply the previous factorial multiplied by some new number. So for six factorial, I would simply take my new number of six and multiply it by the previous factorial which is five. So five factorial now six times five factorial is going to give me a value of 720 because all we did was multiply this previous factorial 120 by six. Now to generalize this a little bit more if I have any number. And I'm simply going to take that number and multiply it by N minus one factorial, just like we saw for six factorial, we took six times six minus one or five factorial. Now this is going to allow us to simplify some factorial expressions a bit easier. So let's go ahead and work through some examples and see exactly how that's going to work for us. Now, looking at our first example here, we have four times three factorial. But using this formula that we just learned, this is really just the same thing as four factorial, which we already know the value of up here. We saw that four factorial was equal to 24. So that's going to be our final answer here that four times three factorial is equal to 24. So let's look at our second example here, we have 100 factorial divided by 99 factorial. And your first instinct here might be to go ahead and start multiplying all of these out, counting down to 1, 100 times 99 times 98 times 97 and so on. But we don't have to do that because we have this formula. So I can actually just rewrite 100 factorial using this formula. And let's see what happens. So 100 factorial is really just 100 times 99 factorial. And if I'm dividing all of that by 99 factorial, well, if this is just going to cancel out, so that 99 factorial on the top and the bottom just completely goes away. Leaving me with a simple answer of 100. No crazy multiplication needed. Let's look at our final example here, here we have one minus one factorial. Now you might be thinking that isn't this just zero factorial? And you're right. It is. But remember I said zero factorial was a bit weird. So let's think about this a little bit deeper using this formula here. This one minus one factorial looks kind of similar to this and minus one factorial. So if and was equal to one and we plugged that into this equation or this formula here, we're going to see that one factorial is equal to one times one minus one factorial. Now, here we see this one minus one factorial, which is exactly what we're looking for here. So let's go ahead and simplify this a bit more. Well, I know that one factorial is just equal to one. So this is really just one and that's equal to I have one times one minus one factorial. Now, one times anything is just that thing. So this leaves me that one is equal to one minus one factorial. And we already have our answer here. That one minus one factorial is equal to one, which tells us that zero factorial is actually just equal to one and we can fill in that final value in our table up here that zero factorial is one. So now that we know what a factorial is and how to simplify different expressions. Let's get some more practice. Thanks for watching and I'll see you in the next one.

2

Problem

Problem

Evaluate the expression.

$\frac{12}{4!}$

A

$\frac14$

B

$\frac12$

C

$2$

D

$3$

3

Problem

Problem

Evaluate the expression.

$\frac{9!}{7!}$

A

2!

B

63

C

72

D

98

4

Problem

Problem

Evaluate the expression.

$\frac{16!}{12!\cdot4!}$

A

0

B

1

C

1,820

D

43,680

5

Problem

Problem

Write the first 4 terms of the sequence $a_{n}=n^2\cdot\left(n-1\right)!$