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 :
\[{{(1+x)}^{21}}+{{(1+x)}^{22}}+....+{{(1+x)}^{30}}\] \[={{(1+x)}^{21}}\left[ \frac{{{(1+x)}^{10}}-1}{(1+x)-1} \right]\]= \[\frac{1}{x}[{{(1+x)}^{31}}-{{(1+x)}^{21}}]\] \ Coefficient of x5 in the given expression = Coefficient of x5 in \[\left\{ \frac{1}{x}[{{(1+x)}^{31}}-{{(1+x)}^{21}}] \right\}\] = Coefficient of x6 in \[[{{(1+x)}^{31}}-{{(1+x)}^{21}}]\] = \[{}^{31}{{C}_{6}}-{}^{21}{{C}_{6}}\].
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