Consider the following statements in respect of the expansion of \((x^2+x^{-2}+2)^4\): I. The number of terms in the expansion is 5. II. One of the coefficients in the expansion has a maximum value equal to 70. Which of the statements given above is/are correct?
II only
Since \(x^2+x^{-2}+2=\left(x+\dfrac{1}{x}\right)^2\), the given expression equals \(\left(x+\dfrac{1}{x}\right)^8\), whose expansion has terms with exponents \(8, 6, 4, \ldots, -8\), i.e. 9 terms, not 5, so statement I is false. The coefficients are \(\binom{8}{k}\) for \(k=0,\ldots,8\), whose maximum value is \(\binom{8}{4}=70\), so statement II is true.
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