Discussion
The coefficient of \(x^{10}\) in \((1 - x^{2})(1 - x^{3})^{9}\) is
Answer: E
The coefficient is 0.
Every term of \((1 - x^{3})^{9}\) has a power of x that is a multiple of 3: 0, 3, 6, 9, ... Multiplying by \(1 - x^{2}\) gives powers of the form 3k (from the 1) or 3k + 2 (from \(-x^{2}\)).
To get \(x^{10}\) we would need 3k = 10 or 3k + 2 = 10, that is k = 10/3 or k = 8/3. Neither is a whole number, so no term contains \(x^{10}\). The binomial coefficients in the other options belong to powers such as \(x^{12}\) or \(x^{14}\), not to \(x^{10}\).
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