A) 605
B) 810
C) 990
D) 910
Correct Answer: C
Solution :
[c] \[\because {{(1+x+{{x}^{2}}+{{x}^{3}})}^{11}}={{(1+x)}^{11}}{{(1+{{x}^{2}})}^{11}}\] \[=(1{{+}^{11}}{{C}_{1}}x{{+}^{11}}{{C}_{2}}{{x}^{2}}{{+}^{11}}{{C}_{3}}{{x}^{3}}{{+}^{11}}{{C}_{4}}{{x}^{4}})+\] ??.)\[\times \{1{{+}^{11}}{{C}_{1}}{{x}^{2}}{{+}^{11}}{{C}_{2}}{{({{x}^{2}})}^{2}}{{+}^{11}}{{C}_{3}}{{({{x}^{2}})}^{3}}+....\}\] Hence, the coefficient of x4 \[{{=}^{11}}{{C}_{2}}{{+}^{11}}{{C}_{4}}{{+}^{11}}{{C}_{2}}{{.}^{11}}{{C}_{1}}=\frac{11}{2.2}+\frac{11}{4.7}+\frac{11}{2.9}\times 11\]\[=\frac{11\times 10}{2}+\frac{11\times 10\times 9\times 8}{4\times 3\times 2}+\frac{11\times 10\times 11}{2}\] \[=55+330+605=990\] Hence, option [c] is correct. \[{{(1+x+{{x}^{2}}+{{x}^{4}})}^{11}}=(1+{{x}^{2}})\]
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