>> A notation that is associated with sequences quite often is called factorial notation. And it can be thought off as simply an operation symbol. Factorial notation is indicated with an exclamation point and I can define it most easily by showing an example. 5 factorial is equal to 1 times 2 times 3 times 4 times 5. So we're beginning with a factor of 1 and we're multiplying times integer values all the way up to the number indicated here. Now the 1 is not gonna play a big role in changing the value here so it could be left out and often we'll leave it out. But just for emphasis, technically it begins at 1 and it goes all the way up to the number indicated here. Now another to list the factors of course is in the reverse order and a lot of times when we use factorial notation, this is a handy way to do it because of cancellation opportunities that we'll see very shortly. Okay, then the idea is to multiply the factors and get some kind of result. Now factorial numbers, when you take the factorial of a number, the value of that number to the factorial idea gets really, really big really, really fast. In fact you might use your calculator and make the calculation of factorial. And I forget at the moment where the factorial symbol is on the calculator. Maybe I'll tell you that later. It's probably in the math menu under number or something like that. But something like 80 factorial which seems to be a relatively smaller number, I believe, has to be listed in scientific notation by the calculator because you run out of digits pretty quickly when you talk about such huge numbers. At any rate let's talk about some other aspects of factorial. As a special definition, 0 factorial is define to be 1. Now here's a technique that's often used when factorials are involved in problems. If have 12 factorial over 10 factorial, you know this is just a string of factors, 12 times 11 times 10 times 9, all the way down to 1. And this is a string of factors. It's 10 times 9 times 8 times 7 and so on. And we could list all of those factors for numerator and denominator and take a look at the cancellation opportunities. Well, we can see them a little more efficiently I think like this. We could rewrite 12 as 12 times 11 times 10 factorial and I'm stopping right there because I noticed that the denominator is 10 factorial and I'll just bring it over. Now this indicates a bunch of factors and that indicates a bunch of factors and they're all the same. So I can cancel these two, you see, without listing all of the separate factors. And now I have just 12 times 11 to evaluate here for the value of this expression. In a similar way, we can perform a cancellation or simplify when letters are involved. N plus 2 factorial over N factorial can be simplified like this. Now I'm gonna come down. The N plus 2 factorial means we're taking the term N plus 2 and then multiplying times, let's see. 1 less than that would be N plus 2 minus 1 or simply N plus 1 and then 1 less than that means N plus 1 minus 1 and that simply N. So this is N. I'll list it as N factorial because in the denominator I have this N factorial and I know that all of these factors involved with this cancel with all of these factors. They're gone and so my expression becomes simply N plus 2 times N plus 1. And that's all there is to it.