Q: A group of 100 students play various instruments: 70 play trombone, 75 play accordion, 85 play violin and 80 play guitar.  What is the minimum number of students who must play all 4?

A: 10 students is the minimum number that must play all 4 instruments

Here we need to look for the overlap since there must be some students who play more than one instrument.  The best way to do this is with a Venn diagram (which was really popular during the New Math of the ’70’s when I was in school!).

The Math below is done to find the minimum number of students who must play all 4:

So if 70 play trombone and 75 play accordion (which add up to 145) there must be at least a 45 overlap that play both.

We will now consider those 45 (that play 2 instruments) with the 85 who play violin (which add up to 130) so there must be a 30 overlap that play 3.

We will now consider those 30 with the 80 who play guitar (which add up to 110) so there must be a 10 overlap that play all 4 instruments.