We need to choose a team of 6 people with at least 1 swimmer, 1 athlete, and 1 cyclist, and more athletes than cyclists. Let's consider the possible distributions:

1. 1 swimmer, 3 athletes, 2 cyclists:

   \(egin{align*} 5 \times \binom{7}{3} \times \binom{4}{2} &= 5 \times 35 \times 6 = 1050 ext{ ways} div>\)

2. 1 swimmer, 4 athletes, 1 cyclist:

   \(egin{align*} 5 \times \binom{7}{4} \times 4 &= 5 \times 35 \times 4 = 700 ext{ ways} div>\)

3. 2 swimmers, 3 athletes, 1 cyclist:

   \(egin{align*} \binom{5}{2} \times \binom{7}{3} \times 4 &= 10 \times 35 \times 4 = 1400 ext{ ways} div>\)

4. 3 swimmers, 2 athletes, 1 cyclist:

   \(egin{align*} \binom{5}{3} \times \binom{7}{2} \times 4 &= 10 \times 21 \times 4 = 840 ext{ ways} div>\)

Adding all these possibilities gives the total number of ways:

\(egin{align*} 1050 + 700 + 1400 + 840 = 3990 ext{ ways} div>\)