Here is the Venn Diagram:
(sorry, you must imagine the three interconnecting circles, but draw them so that Milk includes the overlapping sections with an M, Coffee includes the overlapping sections with C, and Tea includes the overlapping sections with T; None is in the Universe, but outside all the circles):
The Universe = 100
- - - - - - - - - - - - - - - - - - - - - - - -
Most problems like this give the values of the “Only” items first, but require you to solve the problem in the reverse order, so get used to that approach.
“100 students” means The Universe (the grand total)
“10 students had all the three drinks” means M&C&T [note: start at the middle]
Realize that M&C includes M&C&T, so thatt has to be subtracted.
“20 had milk and coffee” means M&C = 20-M&C&T = 20-10 = 10
“30 had coffee and tea” means C&T=30-M&C&T = 30-10 = 20
“25 had milk and tea” means M&T=25-M&C&T = 25-10 = 15
That fills in all of the overlapping sections of the diagram. Now, the “only” sections are given.
“12 had milk only” means MilkOnly = 20
“5 had coffee only” means CoffeeOInly = 5
“8 had tea only” means TeaOnly=8
Now, you have to add up all of the little segments along with the None that is outside of the circles to get the total of 100 in the Universe (the entire group)
O.K., solve for None:
The number of students that did not take any of the three drinks is 20.