A) 256
B) 128
C) 512
D) 64
Correct Answer: C
Solution :
\[{{(1+x+{{x}^{2}}+{{x}^{3}})}^{5}}={{(1+x)}^{5}}{{(1+{{x}^{2}})}^{5}}\]\[=(1+5x+10{{x}^{2}}+10{{x}^{3}}+5{{x}^{4}}+{{x}^{5}})\]\[\times (1+5{{x}^{2}}+10{{x}^{4}}+10{{x}^{6}}+5{{x}^{8}}+{{x}^{10}})\] Therefore the required sum of coefficients\[=(1+10+5){{.2}^{5}}=16\times 32=512\] Note: \[{{2}^{n}}={{2}^{5}}\]= Sum of all the binomial coefficients in the 2nd bracket in which all the powers of x are even.
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