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

$C_8^3=\dfrac{8!}{3!(8-3)!}=\dfrac{8!}{3!\cdot 5!}=\dfrac{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8}{1\cdot 2\cdot 3\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5} =56$ ways.

They can choose 4 flutists from 11 in

$C_{11}^4=\dfrac{11!}{4!(11-4)!}=\dfrac{11!}{4!\cdot 7!}=\dfrac{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11}{1\cdot 2\cdot 3\cdot 4\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7} =330$ ways.

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