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Prove that .^(n)C(1) + 2 xx .^(n)C(2) + ...

Prove that `.^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "…." + n xx .^(n)C_(n) = n2^(n-1)`.
Hence, prove that
`.^(n)C_(1).(.^(n)C_(2))^(2).(.^(n)C_(3))^(3)"......."(.^(n)C_(n))^(n) le ((2^(n))/(n+1))^(.^(n+1)C_(2)) AA n in N`.

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To solve the given problem, we will break it down into two parts as stated in the question. ### Part 1: Prove that \[ \sum_{k=1}^{n} k \cdot \binom{n}{k} = n \cdot 2^{n-1} \] **Step 1: Start with the Binomial Theorem** ...
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