Question

A town orchestra is holding auditions for 3 new flute players and 4 new trumpet players. If8 flutists try out, and 11 trumpeters try out, how many ways can the conductor choose his new members

Answer 1

A town orchestra needs 3 new flute players and 4 new trumpet players. 8 flutists and 11 trumpeters try out, then the orchestra should select 3 flutists from 8 and 4 trumpeters form 11.

They can choose 3 flutists from 8 in

`C_8^3=(8!)/(3!(8-3)!)=(8!)/(3!\cdot 5!)=(1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8)/(1\cdot 2\cdot 3\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5) =56` ways.

They can choose 4 flutists from 11 in

`C_(11)^4=(11!)/(4!(11-4)!)=(11!)/(4!\cdot 7!)=(1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11)/(1\cdot 2\cdot 3\cdot 4\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7) =330` ways.

Use the product rule: 56·330=18480 ways for conductor to choose his new members.