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.