A school band has 43 members. Sixteen play the piano, 17
play the bassoon and 23 play the drums. Five of these students play
both the piano and the bassoon, 11 play both the bassoon and the drums
and 8 play both the piano and the drums. Three students play all three
instruments. How many students play exactly two of these instruments?
Well, it seems like we have a whole orchestra going on here! To find out how many students play exactly two instruments, let's do some math (or math-ical humor, if you will).
First, we need to add up the number of students who play each instrument individually. We have 16 pianists, 17 bassoonists, and 23 drummers. Now, let me put on my goofy hat and calculate.
If we add 16 + 17 + 23, we get 56. However, we need to subtract the students who play multiple instruments (since we've counted them more than once).
There are 5 students who play both piano and bassoon, 11 who play both bassoon and drums, and 8 who play both piano and drums. When we add those together (5 + 11 + 8), we get 24.
But wait, we've counted the 3 students who play all three instruments twice! So let's subtract that from our total (24 - 3 = 21).
Now, let's put on our grand finale with a drumroll! To find out the number of students who play exactly two instruments, we subtract 21 from our initial total of 56.
56 - 21 = 35
Voila! We have 35 students who play exactly two of these instruments. Bravo to the multitasking maestros!
To find the number of students who play exactly two instruments, we can use the principle of inclusion-exclusion.
Let's calculate the number of students who play exactly two instruments:
Number of students who play piano and bassoon = 5
Number of students who play bassoon and drums = 11
Number of students who play piano and drums = 8
Now, let's find the sum of these three groups to get the total number of students who play exactly two instruments:
5 + 11 + 8 = 24
Therefore, there are 24 students who play exactly two instruments.
To find out the number of students who play exactly two instruments, we need to subtract the number of students who play all three instruments from the total number of students who play two or more instruments.
Given information:
Total number of band members (n) = 43
Number of students playing the piano (P) = 16
Number of students playing the bassoon (B) = 17
Number of students playing the drums (D) = 23
Number of students playing both the piano and the bassoon (P ∧ B) = 5
Number of students playing both the bassoon and the drums (B ∧ D) = 11
Number of students playing both the piano and the drums (P ∧ D) = 8
Number of students playing all three instruments (P ∧ B ∧ D) = 3
To find the number of students who play exactly two instruments, we can follow the steps below:
1. Calculate the total number of students playing two or more instruments:
Total students playing two or more instruments = (P ∪ B ∪ D) = (P + B + D - (P ∧ B) - (B ∧ D) - (P ∧ D) + (P ∧ B ∧ D))
Total students playing two or more instruments = (16 + 17 + 23 - 5 - 11 - 8 + 3)
2. Calculate the number of students playing exactly two instruments:
Students playing exactly two instruments = Total students playing two or more instruments - Students playing all three instruments
Students playing exactly two instruments = (16 + 17 + 23 - 5 - 11 - 8 + 3) - 3
Now, let's calculate it:
Students playing exactly two instruments = (16 + 17 + 23 - 5 - 11 - 8 + 3) - 3
Students playing exactly two instruments = 53 - 3
Students playing exactly two instruments = 50
Therefore, there are 50 students who play exactly two instruments in the school band.