Find the sum `.^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "……" + n xx .^(n)C_(n)`.

Explanation

To find the sum \( S = 1 \cdot \binom{n}{1} + 2 \cdot \binom{n}{2} + 3 \cdot \binom{n}{3} + \ldots + n \cdot \binom{n}{n} \), we can use a clever approach involving the binomial theorem.

### Step-by-Step Solution:

1. **Define the Sum**:
Let \( S = 1 \cdot \binom{n}{1} + 2 \cdot \binom{n}{2} + 3 \cdot \binom{n}{3} + \ldots + n \cdot \binom{n}{n} \).

2. **Include \( n \cdot \binom{n}{0} \)**:
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