I have trouble finding an explicit expression to prove this sum using induction, and would like a hint. It would also help if someone provided a non-induction question.
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write down binomial theorem, stare at it. – mdave16 Oct 27 '17 at 23:31
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been there, done that. I can't simplify it down to the binomial theorem or I wouldn't be asking. – Gerard L. Oct 27 '17 at 23:32
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2Hint: $\binom{n}{k} = \frac{n}{k} \binom{n-1}{k-1}$. – Daniel Schepler Oct 27 '17 at 23:33
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2Me thinks you haven't stared enough, you may want to include "no calculus please" in your question – mdave16 Oct 27 '17 at 23:33
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Why the downvote? – Gerard L. Oct 28 '17 at 00:11
2 Answers
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Hint: $$ \binom{n}{k}k = \frac{n!}{k!(n-k)!}k = \frac{n!}{(k-1)!(n-k)!}=n\binom{n-1}{k-1} $$
For the last step, note that $n-k = (n-1)-(k-1)$
Hyperplane
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Hint:
What is the derivative of $(1+x)^n$?
Bernard
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Um, I'd prefer to things that the eight grade me could understand. I would rather not do calculus on this problem. – Gerard L. Oct 27 '17 at 23:32
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2Pardon me, but derivatives and Newton's formula are seen in high school, as far as I know. Now if you don't want a two-line computation, I don't know how to do it. – Bernard Oct 27 '17 at 23:35
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@Gerard L. When I was in 8th grade, I barely knew what exponents where (and by 12th grade most people around me barely knew what's a function contrary to Bernard's post), so you're in much, much better position to understand this than I was. – Maximal Ideal Oct 27 '17 at 23:45