Answer:
[tex]n(S\cap F \cap R)=7[/tex]
Step-by-step explanation:
The Universal Set, n(U)=2092
[tex]n(S)=1232\\n(F)=879\\n(R)=114[/tex]
[tex]n(S\cap R)=23\\n(S\cap F)=103\\n(F\cap R)=14[/tex]
Let the number who take all three subjects, [tex]n(S\cap F \cap R)=x[/tex]
Note that in the Venn Diagram, we have subtracted [tex]n(S\cap F \cap R)=x[/tex] from each of the intersection of two sets.
The next step is to determine the number of students who study only each of the courses.
[tex]n(S\:only)=1232-[103-x+x+23-x]=1106+x\\n(F\: only)=879-[103-x+x+14-x]=762+x\\n(R\:only)=114-[23-x+x+14-x]=77+x[/tex]
These values are substituted in the second Venn diagram
Adding up all the values
2092=[1106+x]+[103-x]+x+[23-x]+[762+x]+[14-x]+[77+x]
2092=2085+x
x=2092-2085
x=7
The number of students who have taken courses in all three subjects, [tex]n(S\cap F \cap R)=7[/tex]
