A) \[{{2}^{7}}\]
B) \[8\ .\ {{2}^{8}}\]
C) \[10\ .\ {{2}^{7}}\]
D) None of these
Correct Answer: C
Solution :
The number of selections = coefficient of \[{{x}^{8}}\]in \[(1+x+{{x}^{2}}+.........+{{x}^{8}})(1+x+{{x}^{2}}+......+{{x}^{8}}).{{(1+x)}^{8}}\]= coefficient of \[{{x}^{8}}\] in \[{{\frac{(1-{{x}^{9}})}{{{(1-x)}^{2}}}}^{2}}{{(1+x)}^{8}}\] = coefficient of \[{{x}^{8}}\] in \[{{(1+x)}^{8}}{{(1-x)}^{-2}}\] = coefficient of \[{{x}^{8}}\] in \[{{(}^{8}}{{C}_{0}}{{+}^{8}}{{C}_{1}}x{{+}^{8}}{{C}_{2}}{{x}^{2}}+.......+{{\,}^{8}}{{C}_{8}}{{x}^{8}})\] \[\times (1+2x+3{{x}^{2}}+4{{x}^{3}}+......+9{{x}^{8}}+.....)\] \[=9\ .{{\ }^{8}}{{C}_{0}}+8\ .{{\ }^{8}}{{C}_{1}}+7\ .{{\ }^{8}}{{C}_{2}}+.........+1\ .{{\ }^{8}}{{C}_{8}}\] \[={{C}_{0}}+2{{C}_{1}}+3{{C}_{2}}+.....+9{{C}_{8}}\] \[[{{C}_{r}}{{=}^{8}}{{C}_{r}}]\] Now \[{{C}_{0}}x+{{C}_{1}}{{x}^{2}}+.......+{{C}_{8}}{{x}^{9}}=x{{(1+x)}^{8}}\] Differentiating with respect to \[x\], we get \[{{C}_{0}}+2{{C}_{1}}x+3{{C}_{2}}{{x}^{2}}+...9{{C}_{8}}{{x}^{8}}={{(1+x)}^{8}}+8x{{(1+x)}^{7}}\] Putting \[x=1,\ \]we get \[{{C}_{0}}+2{{C}_{1}}+3{{C}_{2}}+......+9{{C}_{8}}\] \[={{2}^{8}}+8\ .\ {{2}^{7}}={{2}^{7}}.(2+8)=10\ .\ {{2}^{7}}\].
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