burgerandcheese
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- Jul 2, 2018
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What book?View attachment 10781
This is used to determine the coefficient of the nth row and (r + 1)th column of the Pascal's triangle.
In this book they also used this formula to prove (n, r) = n! /[ r!(n - r)!]
View attachment 10781
This is used to determine the coefficient of the nth row and (r + 1)th column of the Pascal's triangle.
In this book they also used this formula to prove (n, r) = n! /[ r!(n - r)!]
View attachment 10781
This is used to determine the coefficient of the nth row and (r + 1)th column of the Pascal's triangle.
In this book they also used this formula to prove (n, r) = n! /[ r!(n - r)!]
@Dr. PTo answer your question specifically, in order to justify the claim that \(\displaystyle \dbinom{n}{r + 1} = \dfrac{n - r}{r + 1} \dbinom{n}{r}\), we can think about what it means. Suppose we have n items, and already know how many ways there are to choose r of them. How many ways are there to choose r+1? Well, after choosing r, there are n-r other items to choose for the extra one we want, so we can multiply by n-r; but then we could have obtained the same set of r+1 items in r+1 different ways (because any of them might have been the last one we added); so we have to divide by r+1.
There are more natural recurrences I have seen used, but they generally rely on similar (though often simpler) thinking. I expect that something like this is what the book did. But I'll want to see the proof in its entirety (and also what leads up to it) in order to be sure. It depends on their starting point.
To answer your question specifically, in order to justify the claim that \(\displaystyle \dbinom{n}{r + 1} = \dfrac{n - r}{r + 1} \dbinom{n}{r}\), we can think about what it means. Suppose we have n items, and already know how many ways there are to choose r of them. How many ways are there to choose r+1? Well, after choosing r, there are n-r other items to choose for the extra one we want, so we can multiply by n-r; but then we could have obtained the same set of r+1 items in r+1 different ways (because any of them might have been the last one we added); so we have to divide by r+1.
I mean, I want to know the proof for the formula, because the book never gave a proof for it.
This was from A-level Pure Mathematics 1 by Hugh Neill and Douglas Quadling:
http://gcecompilation.com/wp-content/uploads/2017/Pure Mathematics -1.pdf
under the topics sequences and binomial theorem. It was first mentioned at page 122 under 8.4 Pascal Sequences
View attachment 10782
Then, it was used again at the bottom of page 135 under the chapter Binomial Theorem and page 136
View attachment 10783
Actually, I feel it is best to wait to give thanks until you feel that you have been fully answered. The primary reason is that those who give answers want, for various reasons, to know that they have been fully responsive. The only way we know that is if the original poster says something along the lines "Got it now. Thanks."I forgot to say thank you JeffM and Dr.Peterson for your helpful answers! Also, I guess this might affect users because my post will go up and other posts from people who need help may go unnoticed, so sorry! Perhaps I should say thanks in advance next time ~