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Let \(n \in \mathbb{N}^*\).
  1. By differentiating the binomial expansion of \((1+x)^n\), show that: $$ n(1+x)^{n-1} = \sum_{k=1}^{n} k\binom{n}{k}x^{k-1} $$
  2. Hence, deduce that: $$ \sum_{k=1}^{n} k\binom{n}{k} = n2^{n-1} $$

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