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  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:$$ \binom{n}{1}+2\binom{n}{2}+3\binom{n}{3}+\ldots+n\binom{n}{n}=n2^{n-1} $$

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