(i) The probability of performing the routine correctly is 0.65, and incorrectly is 0.35. We use the binomial probability formula:

\(P(X = 5) = (0.65)^5 \times (0.35)^2 \times \binom{7}{5}\)

\(= (0.65)^5 \times (0.35)^2 \times 21\)

\(= 0.298\)

(iii) The expected number of correct performances is given by \(np\), where \(p = 0.65\). We need \(np \geq 8\).

\(0.65n \geq 8\)

\(n \geq \frac{8}{0.65} \approx 12.31\)

The smallest integer \(n\) is 13.