54 students in band, 62 in chorus, 37 in orchestra. 28 in band and chorus, 20 in band and orchestra,17 in chorus and orchestra. 10 in all three.

Question

How many total students are in some sort of musical group

Answer 1

Let's start with all three and work backwards from there. Feel free to build a Venn Diagram as we do.

10 take all three.

17 are in Orchestra and Chorus. 17 - 10 (who're in Band too) = 7 in just the two

28 in Band and Chorus. 28 - 10 (who're in Orch too) = 18 in just those two

20 in Band and Orchestra. 20 - 10 (you get the drill) = 10 in solely those.

 

Now we can subtract the multi-groups to find those in only one:

Band: 54 - 10 - 10 - 18 = 16

Chorus: 62 - 18 - 10 - 7 = 27

Orchestra: 37 - 10 - 10 - 7 = 10

 

And with that, every position in the Venn Diagram should be filled. Add them all together to get your final answer.