Binomial Coefficients proof

Question

I tried various methods, but I don't know how to proceed further, because I am fairly new to this chapter.

Please hep!

Answer 1

If differentiation is not mandatory,

for $n\ge r>0,$ $$r\binom nr=r\cdot\dfrac{n\cdot(n-1)!}{\{n-(r-1)\}!r\cdot (r-1)!}=n\binom{n-1}{r-1}$$

$$\implies\sum_{r=1}^nr\binom nrx^{r-1}=n\sum_{r=1}^n\binom{n-1}{r-1}x^{r-1}=n(1+x)^{n-1}$$