A) \[^{51}{{C}_{5}}\]
B) \[^{9}{{C}_{5}}\]
C) \[^{31}{{C}_{6}}{{-}^{21}}{{C}_{6}}\]
D) \[^{30}{{C}_{5}}{{+}^{20}}{{C}_{5}}\]
Correct Answer: C
Solution :
We have, \[{{(1+x)}^{21}}+{{(1+x)}^{22}}+...+{{(1+x)}^{30}}\] \[={{(1+x)}^{21}}\left\{ \frac{{{(1+x)}^{10}}-1}{(1+x)-1} \right\}\] \[=\frac{1}{x}\{{{(1+x)}^{31}}-{{(1+x)}^{21}}\}\] \[\therefore \]Coefficient of \[{{x}^{5}}\] in the given expansion = Coefficient of \[{{x}^{5}}\] in\[\left[ \frac{1}{x}\{{{(1+x)}^{31}}-{{(1+x)}^{31}}\} \right]\] = Coefficient of \[{{x}^{6}}\] in\[\{{{(1+x)}^{31}}-{{(1+x)}^{21}}\}\] \[{{=}^{31}}{{C}_{6}}{{-}^{21}}{{C}_{6}}\]
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