Original subtitles

Instructor: Hello again.

This is going to be a short optional lecture

explaining factorials.

The notion "n factorial" is used to express the product

of the natural numbers from 1 to n.

This means that n factorial equals 1 X 2 X 3,

all the way up to n.

For instance, 3 factorial is equal to 6

since 1 X 2 X 3 = 6.

Simple enough, right?

For the remainder of the lecture,

we are going to explain some important properties

of factorial mathematics.

Before we get into the more complicated concepts,

you should know that there is one odd characteristic.

Negative numbers don't have a factorial

and zero factorial is equal to one by definition.

All right, let's explore the first property.

For any natural number n,

we know that n factorial equals n - 1 factorial times n.

Similarly, n + 1 factorial equals n factorial times n + 1.

For example, 6 factorial equals 5 factorial times 6.

In the same way, 7 factorial equals 6 factorial times 7.

This notion can be expanded further

to express n + k factorial and n - k factorial.

In mathematical terms, this is equivalent to n + k factorial

equals n factorial times n + 1 times n + 2 and so on,

up to n + k.

Similarly, n - k factorial equals n factorial

over n - k + 1 times n - k + 2,

all the way up to n - k + k, which equals n.

For instance, if n is 5 and k is 2,

then 5 + 2 factorial equals 7 factorial

or 5 factorial X 6 X 7.

And also, 5 - 2 factorial equals 3 factorial

or 5 factorial over 4 X 5.

Okay, great.

An important observation

is that if we have two natural numbers, k and n,

where n is the greater number,

then n factorial over k factorial

equals k + 1 times k + 2, all the way up to n.

Let's look at an example where n is 7 and k is 4,

then 7 factorial over 4 factorial

equals the product of the numbers between 1 and 7

over the product of the numbers between 1 and 4.

We can simplify this by crossing out 1, 2, 3 and 4

since they occur in both parts of the fraction.

Doing so leaves us with 5 X 6 X 7.

Great.

Now you have the tools

that will allow you to handle factorial operations.

In the next video, we are going to explore variations.

Thanks for watching.