Name: Combinatorics: Factorials Explained
Uploaded: 2020-04-13T16:30:04.000Z
Duration: 3 min 52 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Future simple

This is going to be a short optional lecture explaining factorials.

To-infinitive

The notation 'n factorial' is used to express the product of the natural numbers

This means that n factorial equals 1, times 2, times 3, all the way up to n.

For instance, 3 factorial is equal to 6, since: 1, times 2, times 3, equals 6

For the remainder of the lecture, we are going to explain some important properties of factorial

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 1 by definition.

For any natural number n, we know that: n factorial equals, n minus 1, factorial,

times n Similarly, n plus one factorial, equals n

For example, six factorial, equals, five factorial times 6.

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

This notion can be expanded further to express n plus k factorial, and n minus k factorial

In mathematical terms, this is equivalent to:

N plus k factorial, equals, n factorial, times, n plus 1, times, n plus 2, and so on, up to

Similarly, n minus k factorial, equals, n factorial,

over n minus k plus 1, times, n minus k plus 2, all the way up to n minus k plus k, which

For instance, if n is 5 and k is 2, then: Five plus two, equals 7 factorial, or, 5 factorial,

times, 6, times 7, and also: Five minus two, equals 3 factorial, or, 5

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 plus

Let's look in the example where n is 7 factorial, and k is 4.

Then, 7 factorial over 4 factorial equals the product of the numbers between 1 and 7,

We can simplify this by crossing out 1,2,3, and 4 since they occur in both parts of the

Doing so, leaves us with 5 times 6 times 7.

Now you have the tools that would allow you to handle factorial operations.