Question

there are 3 different cable channels namelyy
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Answer 1

Given :  There are different cable channels, namely Airtel, TataSky and Dish TV. In a survey, it was found that

85% of the viewers respond to Dish TV, 20% to TataSky, and

30% to Airtel.

20% of the viewers respond to exactly two channels and

5% to none.

https://brainly.in/question/47163603 for complete Question

To Find :  What percentage of the viewers responded to all three channels ​

Solution:

Total = 100 %

n(A)  = 30 %

n(T) = 20 %

n(D) = 85 %

none = 5 %

20% of the viewers respond to exactly two channels and

Total = n(A) + n(T) + n(D)  - n(A ∩ T)  - n(A∩ D) - n(T ∩ D) + n( A∩T∩D) + none

=> 100 % = 30% + 20% + 85 % - ( n(A ∩ T)  + n(A∩ D) + n(T ∩ D) ) +  n( A∩T∩D) + 5 %

=> ( n(A ∩ T)  + n(A∩ D) + n(T ∩ D) ) -  n( A∩T∩D)  = 40 %  

Exactly two channels  =

(n(A ∩ T)  -  n( A∩T∩D)) + ( n(A∩ D)  -  n( A∩T∩D)) + (n(T ∩ D)  -  n( A∩T∩D))

= ( n(A ∩ T)  + n(A∩ D) + n(T ∩ D) ) - 3 n( A∩T∩D)

( n(A ∩ T)  + n(A∩ D) + n(T ∩ D) ) - 3 n( A∩T∩D) = 20 %

=> n(A ∩ T)  + n(A∩ D) + n(T ∩ D)  =  3 n( A∩T∩D) + 20 %

3 n( A∩T∩D) + 20 %   -  n( A∩T∩D)  = 40 %  

=> 2 n( A∩T∩D) = 20 %

=> n( A∩T∩D) =10 %

viewers responded to all three channels ​ =10 %

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