Q) An advertising agency finds that, of its 200 clients who use Television or Radio or both, 150 use Television. How many use only Radio? A). 150, B). 100, C). 50, D). Data is insufficient
According to formula: $$ n(A \cup B) = n(A) + n(B) - n(A \cap B) $$
If $n(A)$ represent number of client using Television and $n(B)$ represents number of client using Radio. Then we have to find $n(B).$
Given information- $n(A \cup B) = 200$ and $n(A) = 150$, then $$ 200 = 150 + x - n(A \cap B) $$ and still we don't have values of $n(A \cap B) $. So, I answered option D). Data in insufficient but to my surprise answer is C).50. HOW?
Option (c) is correct.
However, you are supposed to assume that of the $200$ people, the number of those people who do not use the TV and do not use the radio is $(0)$. This is the key and it (arguably) is suggested/indicated by :
"...its 200 clients who use Television or Radio or both, ..."
The phrase does not say "...or both, or neither".
Edit
The following edit suggested by subsequent comments:
Normally there are $4$ categories: [1]TV only [2] Radio only [3] TV and Radio both[4] neither TV nor radio. I interpret the problem as intending that the $(200)$ statistic refers to the union of the first $3$ categories only.
Once you accept that, then you infer that the people fall into $3$ categories, and the TV watchers comprise $2$ of the $3$ categories (TV only or TV + radio).
This implies that since what is being interrogated is precisely the 3rd category, radio only, the following computation is valid:
$$200 - 150 = 50.$$
Edit
Just to be clear, re my comment following this response:
It is true but irrelevant that the intersection of TV and radio is unknown. It is irrelevant because that information is not needed to solve the problem. The $(150)$ people who watch TV comprise the disjoint union of those people who only use TV + those people who use TV + radio. This is precisely the information needed to solve the problem.