Two ways to do it.  

1. By Venn diagram
2. By formula

By Venn diagram:

Imagine you have all 22 students on a football field.

First draw a big rectangle to represent, say, the big football field, for
all 22 students to stand on.
 

 
Now on that football field draw a big circle for the 12 students
to stand in who like grape juice and label it G:
 

 
Next draw another big circle overlapping it for the 10 students
to stand in who like orange juice and label it O:
 

 
Now we tell the 10 people who like both juices to stand in the overlapping
part of those two circles.  So we'll write  in there to represent the
10 students in that region.
 
 

and in the main drawing:
 


 
Since there are 12 students who like grape juice, the rest of those 12 
besides the 10 that like both, that is, 12-10, or 2, will have to stand in 
this region, in which we write 2:
 
  
 
and in the main drawing:
 

 
---
 
Since there are 15 students who like orange juice, the rest of those 15 
besides the 10 that like both, that is, 15-10, or 5, will have to 
stand in this region, in which we write 5:
 
`                      `  
 
and in the main drawing:
 

 
Now we have accounted for 2 + 10 + 5 or 17 of the 22 students.  That
leaves 22-17, or 5 students who like neither juice, and so they have 
to stand outside the two circles, like this:



So the answer is 5.

----------------------

We could also have done it with a couple formulas:

N(neither G nor O) = 22 - N(G or O). 
 
N(G or O) = N(G) + N(O) - N(G and O)
    
N(G or O) = 12 + 15 - 10 
 
N(G or O) = 17
 
N(neither G nor O) = 22 - N(G or O)

N(neither G nor O) = 22 - 17

N(neither G nor O) = 5
 
Edwin