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