This is going to be a short optional lecture explaining factorials.
The notation ‘n factorial’ is used to express the product of the natural numbers
from 1 to n.
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
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 1 by definition.
All right.
Let’s explore the first property.
For any natural number n, we know that: n factorial equals, n minus 1, factorial,
times n Similarly, n plus one factorial, equals n
factorial, times, n plus one.
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
n plus k
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
equals n.
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
factorial, over 4, times 5.
Ok.
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 plus
1, times, k plus 2, all the way up to n.
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,
over the product of the numbers 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 times 6 times 7.
Great!
Now you have the tools that would allow you to handle factorial operations.