$C_{1} + 2 C_{2} + 3 C_{3} + 4 C_{4} + ......... + n + 1 n C_{n}$

n 2^{n - 1}

2^{n + 1}

2. 2^{n - 1}

Zero

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C_{1} + 2 C_{2} + 3 C_{3} + 4 C_{4} + ......... + n + 1 n C_{n}

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C_{1} + 2 C_{2} + 3 C_{3} + ...... + n C_{n} = n + 2 \times \frac{n ( n - 1 )}{2 !} + 3 \times \frac{n ( n - 1 ) ( n - 2 )}{3 !} + .... + n \times 1 = n + 2 \times \frac{n ( n - 1 )}{1} + 3 \times \frac{n ( n - 1 )}{6} + .... + n = n + \frac{n ( n - 1 )}{1} + \frac{n ( n - 1 )}{2} + ..... + n = n [1 + \frac{n - 1}{1} + \frac{n ( n - 1 )}{2} + ..... + 1] = n [1 + \frac{n - 1}{1 !} + \frac{n ( n - 1 )}{2 !} + ..... + 1] p u t n - 1 = N = n [1 + \frac{N}{1 !} + \frac{N ( N + 1 )}{2 !} + ..... + 1] = n (^{N} C_{0} +^{N} C_{1} +^{N} C_{2} + .... +^{N} C_{N}) = n 2^{N} = n 2^{n - 1} He n ce, t h e o pt i o n A i s t h e correc t an s w er

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