Discussion
The coefficient of \(x^{9}\) in the expansion of \(\left(4 - \frac{x^{2}}{4}\right)^{12}\) is
Answer: E
The coefficient of \(x^{9}\) is 0.
Every term of \(\left(4 - \frac{x^{2}}{4}\right)^{12}\) has the form \({}^{12}C_{r}\, 4^{12-r}\left(-\frac{x^{2}}{4}\right)^{r}\), which contains \(x^{2r}\). Only even powers of x appear, so there is no \(x^{9}\) term and its coefficient is 0.
The options built from \({}^{12}C_{7}\), \({}^{12}C_{6}\) and \({}^{12}C_{5}\) would need \(2r = 9\), which has no whole-number solution.
The general term contains \(x^{2r}\), so this binomial has only even powers of x; any question about an odd power of x in it has the answer 0.
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