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In a group of 500 persons, 300 take tea, 150 take coffee, 250 take cold drink, 90 take tea and coffee, 110 take tea and cold drink, 80 take coffee and cold drink and 50 take all the three drinks.

(i) Find the number of persons who take none of the three drinks.

(ii) Find the number of persons who take only tea.

(iii) Find the number of persons who take coffee and cold drink but not tea.

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Best answer

Given

n(u) = 500, n(T) = 300, 

n(Co) = 150, n(Cd) = 250, 

n(T ∩ Co) = 90, n(T ∪ Cd) = 110, 

n(Co ∩ Cd) = 80, n(T ∩ Cd ∩ Co) = 50

(i) Number of persons who take none of three drinks 

n(u)n(T ∪ Co ∪ Cd) 

{n(T) = like tea} 

{n(Co) = like Coffee} 

{n(Cd) = like Cold drink} 

n(T ∪ Co ∪ Cd) = n(T)) + n(Co)  n(Cd) − n(T ∩ Co) −n(T ∩ Cd) 

= n(C ∩ Cd) + n(T ∩ Cd ∩ Co] 

= 300 + 150 + 250-90-110-80 + 50 

= 750 – 280 

= 470 

n(u) − n(T ∪ Co ∪ Cd) 

= 500 − 470 

= 30 

∴ Number of persons who take none of three drinks = 30. 

(ii) Number of person who take only tea 

n(T) − n(T ∩ Co) − n(T ∩ Cd) + n(T ∩ Cd ∩ Co) 

= 300 - 90 - 110 + 50 

= 350 - 200 

= 150 

∴ Number of person who take only tea = 150 

(iii) Number of persons who take coffee and cold drink but not tea

n(Co ∩ Cd) − n(T ∩ Co ∩ Cd) 

= 80 – 50 

= 30 

∴ Number of persons who take coffee and cold drink but not tea = 30.

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