To solve this problem, we use the binomial probability formula:

\(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)

where \(n\) is the total number of trials, \(k\) is the number of successful trials, and \(p\) is the probability of success on a single trial.

(i) For equal numbers of large and small bands, we need 10 large and 10 small bands. The probability of a band being large is 0.4, and small is 0.6. Thus,

\(P(10 \text{ large and } 10 \text{ small}) = \binom{20}{10} (0.6)^{10} (0.4)^{10}\)

Calculating this gives:

\(= 0.117\)

(ii) For more than 17 small bands, we consider 18, 19, or 20 small bands. The probability of a band being small is 0.6. Thus,

\(P(18, 19, 20 \text{ small}) = \binom{20}{18} (0.6)^{18} (0.4)^{2} + \binom{20}{19} (0.6)^{19} (0.4)^{1} + (0.6)^{20}\)

Calculating each term:

\(= 0.003087 + 0.000487 + 0.00003635\)

Summing these gives:

\(= 0.00361\)