Question

In a sophomore class of 80 students, 42 students take Spanish, 24 students take French and 4 students take both Spanish and French. How many students in the class are not enrolled in either Spanish or French?
A. 18
B. 14
C. 12
D. 10

Answer 1

Hint: These types of questions are solved with the help of Venn diagrams which help us in explaining the relation between various sets. We calculate the union of students taking Spanish and French using the formula for union of two sets i.e. \[n(A \cup B) = n(A) + n(B) - n(A \cap B)\] and then subtract it from the universal set to find number of students who are not enrolled in either Spanish or French.
* Universal set is the set that contains all the sets.

Complete step by step solution:
Here, the class of total \[80\] students is the Universal set which is denoted by \[U\].
Let , set of students taking Spanish be denoted by \[S\].
The set of students taking French be denoted by \[F\].
Then, the set of students taking both Spanish and French will be the intersection of students taking Spanish and students taking French i.e. \[S \cap F\]
The students in the class which are enrolled in either Spanish or French means the students those who are enrolled in Spanish but not in French and students who are enrolled in French but not in Spanish.
So the set of students in the class which are enrolled in either Spanish or French is denoted by \[S \cup F\].

In the Venn diagram, the shaded area represents \[S \cap F\]
Then, number of students taking Spanish, \[n(S) = 42\]
The number of students taking French, \[n(F) = 24\]
The number of students taking both Spanish and French, \[n(S \cap F) = 4\]
To calculate the number of students in the class which are enrolled in either Spanish or French we will calculate \[n(S \cup F)\].
Since, formula for Union of two sets \[A\] and \[B\] is given by \[n(A \cup B) = n(A) + n(B) - n(A \cap B)\].
Substitute the values in the formula.
\[n(S \cup F) = n(S) + n(F) - n(S \cap F)\]
\[n(S \cup F) = 42 + 24 - 4\]
\[n(S \cup F) = 62\]
Students which have not enrolled in either Spanish or in French are the students which have not enrolled in any of the two subjects.
So, Number of students which have not enrolled in either Spanish or in French are equal to total number of students in the class minus the number of students which have enrolled in either of the two subjects.
\[
   = U - n(S \cup F) \\
   = 80 - 62 \\
   = 18 \\
 \]

Thus, option A is correct.

Note:
Students are very likely to get confused with either/or statement and not either/or statement. If we have to calculate ‘either this or that’ kind of value we always calculate the union, but if we have to calculate ‘not either this or that’ which means ‘neither this nor that’ then we calculate the union and subtract it from the universal set.