Evaluate n!/(n-r)! for n = 20 and r = 3

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Accepted Answer

Hello everybody. How are you doing? All right. Today, today we're going to be looking at this question that's telling us to find out the N factorial divided by open parentheses. N minus R closed parentheses factor where N is equal to 31 and R is equal to two. Now, the answers provided are a 60 B C 930 D 93. Now, before we start, we have to recall what and, and factorial or what factorial means to begin with. So I'll go off to the side and I'll recall. N factorial is equal to N multiplied by open parentheses. N minus one closed parentheses, multiplied by open parentheses and minus two closed parentheses dot dot dot Like a sequence and finally multiplied by one as the final term. So if we apply that to what we have in our, in our problem, we have that we'll plug in the N and the R and we'll have 31 factorial in the numerator divided by open parentheses. 31 minus two closed parentheses. Factorial. Now that will be equal to 31 bacterial divided by 31 minus two is 29. So have divided by 29 factorial and now we can go ahead and expand. However, I'm not gonna be expanding the whole thing and I'll show you why in a second. So we have 31 multiplied by minus one, which is 30 multiplied by 31 minus two, which is 29 so on. But I'll stop here. So I'll have 29 I'll call this 29 factorial because we will continue to go on multiply by 28 27 26 so on. However, I can have 31 multiplied by 30 multiplied by factorial divided by 29 factorial in the, in the denominator. And you might be guessing as to why stop of the 29. Now, if you look, we have a 29 factorial on the numerator and a 29 factorial on the denominator. Now, we can treat this as any other number, any other variable. If we have it in the numerator and the denominator, they can cancel out. So we'll cancel out the 29 factorial in the denominator and the 29 factorial in the numerator. And we'll be left with 31 times 30 or multiplied by 30 which is equal to 930. And that will be our final answer or in this case, answer choice C So just like that, we're able to go from a relatively large factorial into a simple multi multiplication of 31 multiplied by 30. So I hope that was helpful and until next time.

Evaluate n!/(n-r)! for n = 20 and r = 3

Key Concepts

Factorial Notation

Permutation Formula

Substitution and Simplification

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