Question

46. In a colony of 125 members, 70 members watch
Telugu channel, 80 members watch Hindi chan-
nel and 95 watch English channel, 20 watch only
Telugu and Hindi, 35 watch only English and
Hindi and 15 watch only Telugu and English.
How many members watch all the three channels,
if each watches either of the channels? ​

Answer 1

GiveN :-

• In a colony of 125 members, 70 members watch Telugu channel.
• 80 members watch Hindi channel and 95 watch English channel .
• 20 watch only Telugu and Hindi.
• 35 watch only English and Hindi and 15 watch only Telugu and English.

To FinD :-

• The number of people who watch all the three channels .

SolutioN :-

Here we will be using a formula of Set Theory . Let us take there are three sets . Set A , Set B and Set C . Then the relation between the number of elements of three sets , number of elements in union of three sets. Number of elements in intersection of sets taken two at a time is given by ,

$\sf\dashrightarrow \pink{ n( A \cup B \cup C ) = n(A)+n(B)+n(C)- n(A \cap B )-n(C\cap B ) - n(A \cap C ) }$

• Firstly let's make a Venn Diagram for the given situation .For the Venn Diagram refer to the attachment . Now let us take that ,

$\red{\frak{Given}}\begin{cases}\textsf{ Number of English viewers = \textbf{ n(E) }.} \\\textsf{ Number of Hindi viewers = \textbf{ n(H) }.} \\\textsf{ Number of Telgu viewers = \textbf{ n(T) }.} \end{cases}$

Also if we convert the given statements , then

$\to \textsf{ Number of people who watch Hindi and English = n( H \cap E) = \textbf{35} .}$

$\to \textsf{ Number of people who watch Hindi and Telgu = n( H \cap T) = \textbf{20} .}$

$\to \textsf{ Number of people who watch Telgu and English = n( T \cap E) = \textbf{15} .}$

$\to \textsf{ Total Number of people= n( H \cup E \cup T ) = \textbf{175} .}$

$\rule{200}2$

$\red{\bigstar}\underline{\textsf{ Now put on the respective values in stated formula :- }}$

$\sf:\implies n( H \cup E \cup T ) = n(H)+n(E)+n(T)- n(H \cap E )-n(E\cap T ) - n(T\cap A ) \\\\\sf:\implies 175 = 95 + 80 + 70 - 35 - 20-15+n(H\cap E \cap T ) \\\\\sf:\implies n(H\cap E \cap T ) + 150 = 175 \\\\\sf:\implies n(H\cap E \cap T ) = 175 - 150 \\\\\sf:\implies\underset{\blue{\sf Required \ Answer }}{\underbrace{\boxed{\pink{\frak{ n(Hindi\cap English \cap Telgu ) = 25 }}}}}$

$\rule{200}2$