Exercises 89–91 will help you prepare for the material covered in...

Question

Exercises 89–91 will help you prepare for the material covered in the next section. Evaluate n!/(n-r)!r! for n = 8 and r = 3

Accepted Answer

Hello everybody. Everything. All right. Today, today, we're going to be looking at this question that states find out and then they give us a fraction where in the numerator we have and factorial and then we have divided by open parentheses and minus R closed parentheses. Factorial multiplied by R factorial. And all of that will be in the denominator where N equals 16 and R equals nine. Now, the S provided are a 12, B 11,440 C four billion, 151 million, 347,200 and the 518,918,400. Now to continue and go ahead with this question, we have to ask ourselves what does factorial mean? So let's recall, we call that N factorial is equal to N multiplied by open parentheses and minus one closed parentheses, multiplied by open parentheses and minus two closed parentheses dot dot dot And so on with the final turn multiplied by one, that is the rule of factors. So if we apply that to what we have, we have 16 factorial and the numerator because we're plugging in the N and the R divided by sixth open parentheses, 16 minus nine closed parentheses. Factorial multiplied by nine factorial. And that will be equal to factorial divided by seven factorial multiplying nine factorial. And that will be equal to, like we said, we'll have 16. So this is where we apply the factorial rule. We'll have 16, multiplied by 15, multiplied by 14, multiplied by 13, multiplied by 12, multiplied by 11, multiplied by 10. And this is where I'll stop and I'll put multiplied by nine factorial. So I'll stop here. This, if I just put nine factorial, it, this is like saying after the 10, it continues to multiply by eight or, or multiply by 987 and so on. And we'll have all that divided by seven factorial multiplied by nine factorial. And now you may be able to guess why stop at the nine factorial because if we have nine factorial and the numerator and the denominator, this can be applied like any other number, any other variable. And then nine factorial would cancel out and we would be left with, let's so, OK, so let's rewrite what we'd be left with. We would be left with 16, multiplied by 15, multiplied by 14, multiplied by 13, multiplied by 12, multiplied by 11, multiplied by 10. And we stopped there because we canceled out the nine factor and so on. And we have all of that multiplication divided by the seven factorial which is seven, multiplied by six, multiplied by five, multiplied by four, multiplied by three, multiplied by two and finally multiplied by one. Now, when you plug all of that into your calculator, it'll give you a fraction of 57 million, 600 1006 100 divided by 5040. And when you divide that out, it'll simplify to give you 11,440 as your final answer or in this case, answer choice B so I hope that was helpful and until next time.

Exercises 89–91 will help you prepare for the material covered in the next section. Evaluate n!/(n-r)!r! for n = 8 and r = 3

Key Concepts

Factorial

Permutations and Combinations

Binomial Coefficient

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