In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?

Solution

Maths sir · 3 steps

Find where the two smaller circles meet; if that meeting region sits inside the big circle, all three overlap.

1. What we need. Each circle is one game. We need some region shared by exactly two circles, and no region shared by all three circles.
2. How to test. In every diagram, the two smaller circles cross each other. Their lens-shaped overlap is the region of students playing both of those games. If that lens lies inside the big circle, those students play all three games.

   Diagram	Do the small circles overlap?	Where is that overlap?	Region common to all three?
   P	Yes	Fully inside the big circle	Yes, so P fails
   Q	Yes, a thin sliver in the middle	Inside the big circle	Yes, so Q fails
   R	Yes	Inside the big circle	Yes, so R fails

3. So. Each of P, Q and R has a region shared by all three games, which contradicts "none play all three". No diagram fits, so the answer is (b).

The idea to keep. To check that no one plays all three, find the overlap of any two circles and see whether the third circle covers it. Even a thin sliver inside the third circle counts.

Answer from the NTA final answer key. Solution by PrepWiser.