Compute ${n \choose 0}+2{n \choose 1}+3{n \choose 2} + \cdots + (n+1){n \choose n}$

Question

I am having trouble doing this question and I do not know where to start. Any help is appreciated.

Your question should be clear without the title. After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title. — Martin R, Oct 07 '19 at 13:47
Possible duplicate of Simplify $\sum_{i=0}^n (i+1)\binom ni$ – found instantly with Approach0 — Martin R, Oct 07 '19 at 13:48

score 0 · Answer 1 · answered Oct 07 '19 at 13:45

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Hint: Consider $f(x)={n \choose 0}x+{n \choose 1}x^2+{n \choose 2}x^3 + \cdots + {n \choose n}x^{n+1}$. Write $f(x)=xg(x)$.

answered Oct 07 '19 at 13:45

lhf

score 0 · Answer 2 · answered Oct 07 '19 at 13:49

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For $r\ge1,$

$$(r+1)\binom nr=r\binom nr+\binom nr=n\binom{n-1}{r-1}+\binom nr$$ Now use $$(1+1)^m=\sum_{r=0}^m\binom mr$$

answered Oct 07 '19 at 13:49

lab bhattacharjee

2 Answers2