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\( \begin{aligned} \text { If }\left(x+a_{1}\right)\left(x+a_{2}\right)\left(x+a_{3}\right) \ldots\left(x+a_{n}\right)=& x^{n}+S_{1} x^{n-1} \\ &+S_{2} x^{n-2}+\ldots+S_{n} \end{aligned} \)
\( \mathrm{P} \) where, \( S_{1}=\sum_{i=1}^{n} a_{i}, S_{2}=\sum_{1 \leq ij \leq n} a_{i} a_{j}, S_{3}=\sum_{1 \leq ijk \leq n} \sum_{ji} a_{i} a_{j} a_{k} \)

W. and so on.
If \( (1+x)^{n}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots+C_{n} x^{n} \), the coefficient of \( x^{n} \) in the expansion of \( \left(x+C_{0}\right)\left(x+3 C_{1}\right)\left(x+5 C_{2}\right) \ldots\left(x+(2 n+1) C_{n}\right) \), is
(a) \( n \cdot 2^{n} \)
(b) \( (n+1) \cdot 2^{n} \)
(c) \( n \cdot 2^{n+1} \)
(d) \( n \cdot 2^{n}+1 \)
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