Discussion
If the coefficient of \(x^{3}\) in the binomial expansion of \((2 + x)^{n}\) is 160, then the coefficient of \(x^{6}\) in the binomial expansion of \((2 - x^{2})^{n}\) is
Answer: C
The coefficient is \(-160\).
First find \(n\). The \(x^{3}\) term of \((2 + x)^{n}\) is \(\binom{n}{3}2^{n-3}x^{3}\).
\[ \binom{n}{3}2^{n-3} = 160 \Rightarrow n = 6 \]
since \(\binom{6}{3} \times 2^{3} = 20 \times 8 = 160\). In \((2 - x^{2})^{6}\), \(x^{6}\) comes from the term with \((-x^{2})^{3}\):
\[ \begin{aligned} \binom{6}{3}2^{3}(-x^{2})^{3} &= 20 \times 8 \times (-x^{6}) \\ &= -160x^{6} \end{aligned} \]
The option 160 drops the minus sign from \((-1)^{3}\).
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