In Exercises 23–28, evaluate each factorial expression. (2n+1)...

Question

In Exercises 23–28, evaluate each factorial expression. (2n+1)!/(2n)!

Accepted Answer

Hello, today we are going to be simplifying the given factorial expression. The factorial expression given to us is seven N plus one factorial divided by seven N factorial. So how can we start simplifying this factorial expression will recall that the factorial, for example, N factorial is the product of the term N and all the terms that come before it. For example, we can simplify or rewrite N factorial to be N multiplied by a term that is one less than N which is N minus one multiplied by another term that is less than N minus one, which will be N minus two multiplied by all the terms that are continuously less than one than each other. That sounds a bit confusing at first. But this is going to be useful in this problem because it'll help us simplify the numerator and denominator. So let's take a look at the numerator in the numerator, we have seven N plus one factorial. If we wanted to expand seven N plus one factorial, we would have to take the product of the original term which is going to be seven N plus one, then multiplied by a term that is less than seven M plus one. So the next term would be seven N plus one minus one. And the reason why we write it with minus one at the end is because the next term in the product must be one less than seven N plus one. This term will then simplify to just be seven N. So we can rewrite the numerator to be seven N plus one multiplied by seven N factorial. And we don't eliminate the factorial because we don't have a stopping point to seven N plus one. So if we take the simplified form of the numerator and put it back into the expression, we can then rewrite the expression to be seven N plus one multiplied by seven N factorial divided by seven N factorial. Notice that we now have a product of seven and factorial in both the numerator and denominator. This means that we can eliminate the terms from the expression and simplify the expression to just be seven and plus one. There's no further way to simplify seven N plus one. So this will be the simplest form of the factorial expression. With that being said, the solution to this problem is going to be C So I hope this video helps you in understanding 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. (2n+1)!/(2n)!

Key Concepts

Factorial Definition

Properties of Factorials

Ratio of Factorials

Watch next