We need to calculate the number of ways to select 6 bicycles with the given constraints.

Consider the following cases:

1. Select 3 mountain bicycles, 1 racing bicycle, and 2 ordinary bicycles:
   The number of ways is given by:
   \(\binom{7}{3} \times \binom{5}{1} \times \binom{8}{2} = 35 \times 5 \times 28 = 4900\).
2. Select 3 mountain bicycles, 2 racing bicycles, and 1 ordinary bicycle:
   The number of ways is given by:
   \(\binom{7}{3} \times \binom{5}{2} \times \binom{8}{1} = 35 \times 10 \times 8 = 2800\).
3. Select 2 mountain bicycles, 2 racing bicycles, and 2 ordinary bicycles:
   The number of ways is given by:
   \(\binom{7}{2} \times \binom{5}{2} \times \binom{8}{2} = 21 \times 10 \times 28 = 5880\).

Adding all these possibilities gives the total number of selections:
\(4900 + 2800 + 5880 = 13,580\).