Question:
The coefficient of $x^{9}$ in the expansion of $(4-\frac{x^{2}}{4})^{12}$ is ________.
The coefficient of $x^{9}$ in the expansion of $(4-\frac{x^{2}}{4})^{12}$ is ________.
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If only $x^2$ is present, all odd power coefficients are zero.
Updated On: Jun 26, 2026
- $-^{12}C_{7}(4)^{7}(3)^{5}$
- $^{12}C_{7}(4)^{7}(3)^{5}$
- $^{12}C_{6}(4)^{6}(3)^{6}$
- $^{12}C_{5}(4)^{5}(3)^{7}$
- 0
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The Correct Option is
Solution and Explanation
Step 1: Concept
The general term is $T_{r+1} = ^{n}C_{r} a^{n-r} b^r$.
Step 2: Meaning
Here $a=4$ and $b=-\frac{x^2}{4}$. The power of $x$ is always $2r$ (even).
Step 3: Analysis
Since $x$ only appears as $x^2$, the expansion only contains even powers of $x$ (0, 2, 4, ..., 24).
Step 4: Conclusion
$x^9$ is an odd power, so its coefficient must be 0. Final Answer: (E)
The general term is $T_{r+1} = ^{n}C_{r} a^{n-r} b^r$.
Step 2: Meaning
Here $a=4$ and $b=-\frac{x^2}{4}$. The power of $x$ is always $2r$ (even).
Step 3: Analysis
Since $x$ only appears as $x^2$, the expansion only contains even powers of $x$ (0, 2, 4, ..., 24).
Step 4: Conclusion
$x^9$ is an odd power, so its coefficient must be 0. Final Answer: (E)
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