In Exercises 23–28, evaluate each factorial expression. 18!/16...

Question

In Exercises 23–28, evaluate each factorial expression. 18!/16!

Accepted Answer

Hello, today we are going to be simplifying the given factorial expression before we jump into this problem. Let's quickly re review how we can simplify factorials. Let's suppose that we are given the general factorial N factorial. A factorial is equivalent to the term given in the factorial multiplied by all the terms that are less than one of itself. So for example, we could rewrite N factorial as N multiplied by a term that's one less than N so N minus one multiplied by another term that is less than N minus one, which will be N minus two and continue writing terms in the products. For a simpler example, let's suppose that we have five factorial, we can rewrite five factorial as the product of five and all the previous terms that come before it leading to one. So we can expand five factorial as five multiplied by four, multiplied by three, multiplied by two, multiplied by one. And this is the kind of trick that we're going to want to use in order to evaluate the given factorial. So the factorial expression given to us is 22 factorial divided by 18 factorial in the numerator we could simplify 22 factorial to be the multiple of 22 and all the terms that come before it to one and do the same for 18. But that process will be quite tedious and lengthy to do. There is a trick that we can do with the numerator 22 factorial can be expanded to be 22 multiplied by 21 multiplied by 20 multiplied by 19, multiplied by 18 factorial. We can rewrite the factorial in this term or in this way because it's the equivalent statement as to writing out all of the terms that come before 22. So instead of expanding 22 completely, we can stop the expansion at 18 factorial. And if we put this back into the factorial expression, we can rewrite the expression as 22 multiplied by 21 multiplied by 20 multiplied by 19, multiplied by 18 factorial divided by 18 factorial. The usefulness of writing 22 factorial in this way is the fact that we have a multiple of 18 factorial in both the numerator and denominator. So what we can do is we can eliminate these terms from the expression. And what we will be left with is the product of 22 21 2019. So now all we have to do is we have to take the product of these four terms. And with the help of a calculator, the product of these 44 terms will be 175,560. And this is going to be the simplest form of the given factorial expression. And with that being said, the answer to this problem is going to be b so I hope this video helps you in understanding on how to work with factorials. And I'll go ahead and see you all in the next video.

In Exercises 23–28, evaluate each factorial expression. 18!/16!

Key Concepts

Factorial Definition

Simplifying Factorials

Properties of Factorials

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