A) \[{{14.2}^{14}}\]
B) \[{{13.2}^{14}}+1\]
C) \[{{13.2}^{14}}-1\]
D) None of these
Correct Answer: B
Solution :
We have \[{{(1+x)}^{15}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}.+....+{{C}_{15}}{{x}^{15}}\] Þ \[\frac{{{(1+x)}^{15}}-1}{x}={{C}_{1}}+{{C}_{2}}x+....+{{C}_{15}}{{x}^{14}}\] Differentiating both sides with respect to x, we get \[=\frac{x.15{{(1+x)}^{14}}-{{(1+x)}^{15}}+1}{{{x}^{2}}}\] = \[{{C}_{2}}+2{{C}_{3}}x+......+{{\,}^{14}}{{C}_{15}}{{x}^{13}}\] Putting\[x=1\], we get \[{{C}_{2}}+2{{C}_{3}}+....+14{{C}_{15}}={{15.2}^{14}}-{{2}^{15}}+1={{13.2}^{14}}+1.\]
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