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

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Accepted Answer

Hello. Everyone in this video. We're going to be looking at this practice problem where we want to simplify the factorial, N plus four factorial divided by n plus five factorial. So I'm going to write the factorial expression off to the side here where I have N plus four factorial in my numerator and N plus five factorial in my denominator. So recall that a factorial is a product of an integer and all the integers below it. So that means that N Plus four factorial is the product of N Plus four times. And We're going to go below four by subtracting one. So N plus four times N plus three and again subtracting one and plus two. And I'm just gonna write another factorial there. So you can see that my end plus four here no longer has a factorial. But when I write and plus four factorial that is the same as writing n plus four times and plus three times N plus two factorial. So I'm gonna use this to help me simplify the expression that we have here more specifically I'm gonna use it for my denominator. So I'm going to keep the numerator as N plus four factorial. But I'm going to write the denominator as N plus five times and plus four factorial and notice how end plus five factorial is still the same thing as N plus five times N plus four factorial because I'm still multiplying by and plus five and all the integers below it by including the N plus four factorial. However from this expression you can see that I have and plus four factorial in both the numerator and denominator of this fraction. So these cancel out to one, So I'm left with one over and plus five. And from here there's really not much else I can do to simplify. So this becomes my final simplification for the factorial expression. And you can see that this aligns with answer choice C. So hopefully this radio hopped and see you next time.

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

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