Question

In a survey of 100 students, the number of students studying the various languages were found to be : English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find:
(i) How many students were studying Hindi?
(ii) How many students were studying English and Hindi? 

Answer 1

Let E, H and S be the sets of students who study English, Hindi and Sanskrit, respectively.
Also, let U be the universal set.
Now, we have:
n(E) = 26, n(S) = 48, n(E\[\cap\]S) = 8 and n(S\[\cap\]H) = 8
Also,
n(E\[\cap\]H\['\]) = 23

\[\Rightarrow\]n(E)\[-\]n(E\[\cap\]H) = 23

\[\Rightarrow\]26 \[-\]n(E\[\cap\]H) = 23

\[\Rightarrow\]n(E\[\cap\]H) = 3

Therefore, the number of students studying English and Hindi is 3 

n(E\[\cap\]H\['\]\[\cap\]S\['\]= 18

\[\Rightarrow\]n(E) \[-\]n{E\[\cap\](H\[\cup\]S)'} = 18

\[\Rightarrow\]26 \[-\]n{(E\[\cap\]H)\[\cup\](E\[\cap\]S)}= 18

\[\Rightarrow\]26 \[-\] {3 + 8 \[-\] n(E\[\cap\]H\[\cap\]S)} = 18

\[\Rightarrow\]n(E\[\cap\]H\[\cap\]S) = 3

Also,
n(E\['\]\[\cap\]H\['\]\[\cap\]S\['\]) = 24

\[\Rightarrow\]n(U) \[-\]n(E\[\cup\]H\[\cup\]S) = 24

\[\Rightarrow\]\[\cup\]H\[\cup\]S) = 76

∴ Number of students studying Hindi = n(E\[\cup\]H\[\cup\]S) \[-\]n(E) \[-\]n(S) + n(E\[\cap\]H) + n(E\[\cap\]S) + n(S\[\cap\]H) \[-\]n(E\[\cap\]H\[\cap\]S) = 76 

\[-\]24 \[-\]48 + 3 + 8 + 8 

\[-\]3
= 18