A) \[{{\,}^{n}}{{C}_{4}}\]
B) \[{{\,}^{n}}{{C}_{4}}+{{\,}^{n}}{{C}_{2}}\]
C) \[{{\,}^{n}}{{C}_{4}}+{{\,}^{n}}{{C}_{4}}+{{\,}^{n}}{{C}_{2}}\]
D) \[{{\,}^{n}}{{C}_{4}}+{{\,}^{n}}{{C}_{2}}^{n}{{C}_{1}}+{{\,}^{n}}{{C}_{2}}\]
Correct Answer: D
Solution :
\[{{(1+x+{{x}^{2}}+{{x}^{3}})}^{n}}\] \[={{[(1+x)+{{x}^{2}}(1+x)]}^{n}}\] \[={{(1+x)}^{n}}{{(1+{{x}^{2}})}^{n}}\] \[=(1+{{\,}^{n}}{{C}_{1}}x{{+}^{n}}{{C}_{2}}{{x}^{2}}{{+}^{n}}{{C}_{3}}{{x}^{3}}{{+}^{n}}{{C}_{4}}{{x}^{4}}+...)\] \[\times \,(1{{+}^{n}}{{C}_{1}}{{x}^{2}}{{+}^{n}}{{C}_{2}}{{x}^{4}}+...)\] \[\therefore \]Coefficient of \[{{x}^{4}}\] \[={{\,}^{n}}{{C}_{2}}{{+}^{n}}{{C}_{1}}^{n}{{C}_{2}}{{+}^{n}}{{C}_{4}}\]
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