Question:

Using the binomial expansion of $(1+x)^n$ and differentiating both sides with respect to $x$, we get $n(1+x)^{n-1} = \sum_{r=1}^{n} r C_r x^{r-1}$. By multiplying by $x$ and applying appropriate operations, combined with the condition that $(r+1)C_r = ^nC_{r+1}(n+2-r)$, we find $n = 17$.