SOLUTION: A high school of 350 students offers biology and chemistry courses. 213 students are enrolled in biology while 155 students are studying chemistry. 78 students are studying both su

There are three ways to do the problem:  Venn diagram method, 
Chart method, and formula method:

Venn diagram method:

Draw a big rectangle to say, represent a big football field.
Assume we start with all 350 students seated in the stands. 



Now draw a big red circle on the football field for all the
biology students to come stand in.  Label the circle B for
"biology"




Next draw a big green circle overlapping the red one for all the
chemistry students to come stand in.  Label the green circle C for
"chemistry"



Next notice that there are 4 regions for the students to come down
and stand in.

I will begin by labeling them I, II, III and Iv

Now we are going to ask all the students to come down from the stands and
stand in which of the four regions they belong in depending of what
course or courses they study or don't study: 



Let's identify each of those four regions. 

Region I:
The students who will stand in region I are those who study biology but
who do not study chemistry.  Notice that they will be inside the red circle
but outside the green circle.

Region II:
The students who will stand in region II are those who study both biology and
chemistry.  Notice that they will be inside the red circle and also
inside the green circle.

Region III:
The students who will stand in region III are those who study chemistry but
who do not study biology.  Notice that they will be inside the green circle
but outside the red circle.

Region IV:
The students who will stand in region IV are those who study neither biology
nor chemistry.  Notice that they will be outside both circles

The object of the problem is to determine how many are in region IV.

We begin by getting the students who are taking both subjects to stand
in region II.  We are given how many will stand in region II by these 
words:

>>...78 students are studying both subjects...<<

Those 78 students will all come down and stand in region II since they must
stand in both circles.  So we'll write 78 where the II is to represent the
78 students that are in both circles because they take both courses:



Next we will fill in region I, which is the rest of the red circle.
These words:

>>...213 students are enrolled in biology...<<

tell us that 213 students must stand in the red circle.  We have already
accounted for 78 of them, so that leaves 213-78 or 135 students to stand in
region I.  So we'll write 135 where the I is to represent the
135 students that are in the red circles that are not in the green circle.



Next we will fill in region III, which is the rest of the green circle.
These words:

>>...155 students are studying chemistry...<<

tell us that 155 students must stand in the green circle.  We have already
accounted for 78 of them, so that leaves 155-78 or 77 students to stand in
region III.  So we'll write 77 where the III is to represent the
78 students that are in the green circle that are not in the red circle.



Now finally we'll tell all the rest of the students to stand in region IV.
That's the region where all the students who do not take either biology
or chemistry will stand.

We know that there are 350 students altogether.  We have accounted for 
135+78+77 or 290 students who take biology or chemistry.  So to find
out how many will stand outside both circles in region IV, we subtract
350 - 290 and get 60.   So the remaining 60 will stand in region IV, outside
both circles.



So the answer is 60.

Chart method (self-explanator if you understand the Venn diagram:


                   Taking Chemistry
                    Yes          No        Totals
          -----------------------------------------   
                |          |            || 
Taking     YES  |   78     |    135     ||  213
Biology   ------|----------|------------||--------- 
            NO  |   77     |     60     ||  137
                |          |            ||
          ------|-----------------------||---------
          ------------------------------||--------- 
Totals          |  155     |    195     ||  350


Formula method

N(B or C) = N(B) + N(C) - N(B and C)

N(B or C) = 213  + 155  -    78 

N(B or C) = 290

N(Neither B nor C) = N(Universal set) - N(B or C)

N(Neither B nor C) =       350        -    290

N(Neither B nor C) = 60

Edwin