(a) To choose a team of 5 people with exactly one adult, select 1 adult from 5 and 4 others from the remaining 7 (3 boys and 4 girls):

\(\binom{5}{1} \times \binom{7}{4} = 5 \times 35 = 175\)

(b) Consider the cases where the team includes at least 2 boys and at least 1 girl:

1. 2 boys, 1 girl, 2 adults: \(\binom{3}{2} \times \binom{4}{1} \times \binom{5}{2} = 3 \times 4 \times 10 = 120\)
2. 2 boys, 2 girls, 1 adult: \(\binom{3}{2} \times \binom{4}{2} \times \binom{5}{1} = 3 \times 6 \times 5 = 90\)
3. 2 boys, 3 girls: \(\binom{3}{2} \times \binom{4}{3} = 3 \times 4 = 12\)
4. 3 boys, 1 girl, 1 adult: \(\binom{3}{3} \times \binom{4}{1} \times \binom{5}{1} = 1 \times 4 \times 5 = 20\)
5. 3 boys, 2 girls: \(\binom{3}{3} \times \binom{4}{2} = 1 \times 6 = 6\)

Summing these cases gives the total number of ways:

\(120 + 90 + 12 + 20 + 6 = 248\)