A) \[C_{0}^{2}+2C_{1}^{2}+3C_{2}^{2}+...+(n+1)C_{n}^{2}\]
B) \[{{C}_{1}}+{{C}_{2}}+{{C}_{3}}+....+{{C}_{n}}\]
C) \[C_{0}^{2}+C_{1}^{2}+C_{2}^{2}+....+C_{n}^{2}\]
D) \[{{C}_{1}}+2{{C}_{2}}+3{{C}_{3}}+....+n{{C}_{n}}\]
E) none of the above
Correct Answer: C
Solution :
\[{{(1+x)}^{n}}{{\left[ 1+\frac{1}{x} \right]}^{n}}\] \[=({{C}_{0}}+{{C}_{1}}x+...+{{C}_{n}}{{x}^{n}})\] \[\times \left( {{C}_{0}}+\frac{{{C}_{1}}}{x}+\frac{{{C}_{2}}}{{{x}^{2}}}+.....+\frac{{{C}_{n}}}{{{x}^{n}}} \right)\] So term independent of\[x\]is \[C_{0}^{2}+C_{1}^{2}+C_{2}^{2}+....+C_{n}^{2}\]
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