A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If $$x$$ denotes the percentage of them, who like both coffee and tea, then $$x$$ cannot be:

We are told that out of all employees in an office, $$73\%$$ like coffee and $$65\%$$ like tea. Let us denote by $$x\%$$ the employees who like both coffee and tea.

To connect these three percentages, we recall the Principle of Inclusion-Exclusion for two sets. For any two sets $$A$$ and $$B$$, it states

$$|A\cup B|=|A|+|B|-|A\cap B|.$$

In our context,

$$$|A|=73\%, \qquad |B|=65\%, \qquad |A\cap B|=x\%.$$$

Hence the percentage of employees who like at least one of the two beverages is

$$$|A\cup B| = 73 + 65 - x = 138 - x\;(\%).$$$

This quantity obviously cannot exceed the total population, which is $$100\%.$$ Therefore, we must have

$$138 - x \le 100.$$

Solving this simple linear inequality step by step, we move all the terms involving $$x$$ to one side:

$$$138 - x \le 100 \\ \Rightarrow -x \le 100 - 138 \\ \Rightarrow -x \le -38.$$$

Now dividing both sides by $$-1$$ (and remembering to reverse the inequality sign), we get

$$x \ge 38.$$

This is our lower bound: the overlap $$x\%$$ must be at least $$38\%.$$

Next, the overlap cannot exceed either of the individual percentages, because the intersection of two sets can never be larger than each set alone. Thus we have two more inequalities:

$$x \le 73 \quad\text{and}\quad x \le 65.$$

The tighter of these two upper bounds is $$65\%,$$ so altogether we have the admissible range

$$38 \le x \le 65.$$

Now we examine the four candidate values:

$$$\begin{aligned} \text{Option A: }&63 &&\text{lies between }38\text{ and }65\ (\text{allowed}),\\ \text{Option B: }&36 &&\text{is }<38\ (\text{not allowed}),\\ \text{Option C: }&54 &&\text{lies between }38\text{ and }65\ (\text{allowed}),\\ \text{Option D: }&38 &&\text{equals the lower bound }38\ (\text{allowed}).\\ \end{aligned}$$$

Thus the only percentage that violates the necessary condition is $$36\%.$$

Hence, the correct answer is Option B.