(i) The probability that Annabel eats pasta on exactly 2 days in a week of 7 days can be modeled using the binomial distribution. The probability of eating pasta on a given day is 0.1, and not eating pasta is 0.9. We use the binomial formula:

\(P(2) = \binom{7}{2} (0.1)^2 (0.9)^5\)

Calculating this gives:

\(P(2) = 21 \times 0.01 \times 0.59049 = 0.124\)

(ii) For the second part, we need to find the probability that Annabel eats rice on 2 days, potato on 1 day, and pasta on 2 days in a period of 5 days. The probabilities are 0.75 for rice, 0.15 for potato, and 0.1 for pasta. The number of ways to arrange these days is given by:

\(\frac{5!}{2!1!2!}\)

The probability is then:

\((0.75)^2 (0.15)^1 (0.1)^2 \times \frac{5!}{2!1!2!}\)

Calculating this gives:

\(0.5625 \times 0.15 \times 0.01 \times 30 = 0.0253\)