A) \[\frac{n(n+1)}{2}\]
B) \[\frac{(n+1)}{n!}\]
C) \[\frac{n({{n}^{2}}+1)}{2}\]
D) \[n\frac{(n+1)}{2}\]
Correct Answer: D
Solution :
Given, \[{{(1+x)}^{n}}{{=}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{2}}{{x}^{2}}+...{{+}^{n}}{{C}_{n}}{{x}^{n}}\] Let\[{{S}_{n}}=\frac{^{n}{{C}_{1}}}{^{n}{{C}_{0}}}+\frac{{{2}^{n}}{{C}_{2}}}{^{n}{{C}_{1}}}+\frac{{{3}^{n}}{{C}_{3}}}{^{n}{{C}_{2}}}+...+\frac{{{n}^{n}}{{C}_{n}}}{^{n}{{C}_{n-1}}}\] Put n= 1,2,3,... \[{{S}_{1}}=\frac{^{1}{{C}_{1}}}{^{1}{{C}_{0}}}=1\] \[{{S}_{2}}=\frac{^{2}{{C}_{1}}}{^{2}{{C}_{0}}}+2.\frac{^{2}{{C}_{2}}}{^{2}{{C}_{1}}}\] \[=\frac{2}{1}+2.\frac{1}{2}=3\] Only option ,\[\frac{n(n+1)}{2}\]satisfies this condition.
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