To perform during the improvisation section of their concert, the director of the school of jazz band draws lots to randomly choose 4 players from the 4 trumpet players, 5 saxophone players, and 2 keyboard players. What is the probability he picks 2 trumpet players and 2 sax players?

1 answer

First, we need to calculate the total number of ways the director can choose 4 players from the available options:

Total number of players = 4 trumpet players + 5 saxophone players + 2 keyboard players = 11 players

Number of ways to choose 4 players from 11 players = 11 choose 4 = 11! / (4!(11-4)!) = 330 ways

Next, we need to calculate the number of ways he can choose 2 trumpet players and 2 saxophone players:

Number of ways to choose 2 trumpet players from 4 = 4 choose 2 = 4! / (2!(4-2)!) = 6 ways

Number of ways to choose 2 saxophone players from 5 = 5 choose 2 = 5! / (2!(5-2)!) = 10 ways

Therefore, the number of ways to choose 2 trumpet players and 2 saxophone players = 6 * 10 = 60 ways

Finally, we calculate the probability by dividing the number of favorable outcomes by the total number of outcomes:

Probability = Number of ways to choose 2 trumpet players and 2 saxophone players / Total number of ways to choose 4 players
Probability = 60 / 330
Probability = 0.1818 or 18.18%