We need to consider different scenarios based on the given conditions:

1. 4 Violinists (V), 1 Cellist (C), and 1 Double Bass player (DB):

   The number of ways to choose 4 violinists from 11 is given by \(\binom{11}{4}\).

   The number of ways to choose 1 cellist from 5 is given by \(\binom{5}{1}\).

   The number of ways to choose 1 double bass player from 4 is given by \(\binom{4}{1}\).

   Total ways for this scenario: \(\binom{11}{4} \times \binom{5}{1} \times \binom{4}{1} = 6600\).

2. 4 Violinists and 2 Cellists:

   The number of ways to choose 4 violinists from 11 is \(\binom{11}{4}\).

   The number of ways to choose 2 cellists from 5 is \(\binom{5}{2}\).

   Total ways for this scenario: \(\binom{11}{4} \times \binom{5}{2} = 3300\).

3. 5 Violinists and 1 Cellist:

   The number of ways to choose 5 violinists from 11 is \(\binom{11}{5}\).

   The number of ways to choose 1 cellist from 5 is \(\binom{5}{1}\).

   Total ways for this scenario: \(\binom{11}{5} \times \binom{5}{1} = 2310\).

Adding all scenarios together gives the total number of ways:

\(6600 + 3300 + 2310 = 12210\).