TS EAMCET · Maths · Binomial Theorem
If \(\left(1+x+x^2+x^3\right)^5=\sum_{k=0}^{15} a_k x^k\), then \(\sum_{k=0}^7 a_{2 k}\) is equal to
- A \(128\)
- B \(256\)
- C \(512\)
- D \(1024\)
Answer & Solution
Correct Answer
(C) \(512\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { Given, }\left(1+x+x^2+x^3\right)^5=\sum_{k=0}^{15} a_k x^k \\ & \Rightarrow[(1+x)+x(1+x)]^5=\sum_{k=0}^{15} a_k x^k \\ & \Rightarrow(1+x)^{10}=a_0 x^0+a_1 x+a_2 x^2+\ldots+a_{15} x^{15} \\ & \Rightarrow \quad{ }^{10} C_0+{ }^{10} C_1 x+{ }^{10} C_2…
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