A survey is done among a population of 200 people who like either tea or coffee. It is found that 60% of the pop lation like tea and 72% of the population like coffee. Let \(x\) be the number of people who like both tea & coffee. Let \(m{\leq x\leq n}\), then choose the correct option.
Suppose the population who like tea is \( n(T) \) and the population who like coffee is \( n(C) \). The population who like either tea or coffee is \( n(T \cup C) \leq 200 \). The population who like tea is: \[ n(T) = \frac{60}{100} \times 200 = 120. \] The population who like coffee is: \[ n(C) = \frac{72}{100} \times 200 = 144. \] Using the principle of inclusion and exclusion, we have: \[ n(T \cup C) = n(T) + n(C) - n(T \cap C). \] Substituting the values: \[ n(T \cup C) = 120 + 144 - n(T \cap C), \] \[ n(T \cap C) = 264 - n(T \cup C). \] As \( 144 \leq n(T \cup C) \leq 200 \), the maximum and minimum values of \( n(T \cap C) \) are \( 120 \) and \( 64 \), respectively. Hence, the difference in the values is: \[ 120 - 64 = 56. \]
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