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There was a previous problem in my homework that basically demonstrated that:

10C7 = 9C6 + 8C6 + 7C6 + 6C6

And our question is: 

"Use that fact to derive a summation formula involving expressions nC1."

I'm not entirely sure what this means, but I'm assuming we are to use Sigma. This is what I came up with:

$${}_nC_r = \sum_{i=r - 1}^{n-1} {}_iC_{r-1}$$

I'm not sure if I'm even using legal notation here, so any help would be greatly appreciated.

JosephTLyons
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  • As an aside, your quoted equality is incorrect. You are off by one as a result of missing the term + 6C6 on the right side. – JMoravitz Sep 20 '17 at 03:34
  • Oops, I forgot to include that term, yes, you are correct. – JosephTLyons Sep 20 '17 at 03:35
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    As an additional aside, it is much more common to notate binomial coefficients as \binom{n}{r} which appears as $\binom{n}{r}$ so long as you have access to any sort of typesetting like we do here. Visit this page to learn more about how to type with MathJax and $\LaTeX$. – JMoravitz Sep 20 '17 at 03:36
  • I'm familiar with this notation, just didn't realize that one was more common than the other. I'll edit my question to reflect this. – JosephTLyons Sep 20 '17 at 03:40
  • @joe_04_04 If you prefer to stick with the written C, you could use ${}_nC_r$ ${}_nC_r$ or ${}_n\mathrm{C}_r$ ${}_n\mathrm{C}_r$ – gen-ℤ ready to perish Sep 20 '17 at 05:04

1 Answers1

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You can use this formula and try and prove the question, $$\binom{n}{r}+\binom{n}{r-1}=\binom{n+1}{r}$$ Where $nCr=\binom{n}{r}$

$$\sum_{i=r-1}^{n-1}\binom{i}{r}$$$$$$$$=\binom{r-1}{r-1}+\binom{r}{r-1}+\binom{r+1}{r-1}+\cdots$$$$$$as $\binom{r-1}{r-1}=\binom{r}{r}=1$$$$$$$\binom{r}{r}+\binom{r}{r-1}+\binom{r+1}{r-1}+\cdots$$$$$$$$\binom{r+1}{r}+\binom{r+1}{r-1}+\binom{r+2}{r-1}+\cdots$$$$$$$$\binom{r+2}{r}+\binom{r+2}{r-1}+\binom{r+3}{r-1}+\cdots$$$$$$$$\binom{r+3}{r}+\binom{r+3}{r-1}+\binom{r+4}{r-1}+\cdots$$$$$$$$\vdots$$$$$$$$\binom{n-1}{r}+\binom{n-1}{r-1}$$$$\binom{n}{r}$$

neonpokharkar
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  • There are much simpler ways to prove the generalization than just repeated application of pascal's identity. See the question I linked to in the above comments. – JMoravitz Sep 20 '17 at 03:40
  • But it is easier to understand – neonpokharkar Sep 20 '17 at 03:51
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    Easier to understand than the following? "To select $r$ numbers from ${0,1,2,3,\dots,n-1}$ you can break into cases based on the largest number selected noting that if the largest number selected was $k$ you could select the remaining numbers in $\binom{k}{r-1}$ ways. Ranging over all possible values of $k$ gives the result" – JMoravitz Sep 20 '17 at 03:55