In an office, every employee likes at least one of tea, coffee and milk. The number of employees who like only tea, only coffee, only milk and all the three are all equal to $x$ . The number of employees who like only tea and coffee, only coffee and milk and only tea and milk are equal and each is equal to the number of employees who like all the three is equal to $x$ . Then the possible value of the number of employees in the office is

65

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90

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77

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85

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Solution

The correct option is C $77$
Let the number people who like all the three be denoted by x.
Therefore
$A \cup B \cup C = n A + n B + n C - n (A \cap B) - n (A \cap C) - n (B \cap C) + A \cap B \cap C$
$= 4 x + 4 x + 4 x - 2 x - 2 x - 2 x + x$
$= 12 x - 6 x + x$
$= 7 x$
Therefore, the total number of people is divisible by 7 which we can see that only true in the case of $77$ .
Hence, option 'C' is correct.

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