In the diagram given below, there is a circle, a square and a triangle dividing the region into five disjoint bounded areas. Each of these areas are labelled with number of players belonging to that area. The circle contains cricketers, the square contains football players and the triangle contains hockey players.
Which one of the following is not correct?
Statement C is “There are some hockey players who play both cricket and football”. This statement is true if the region representing players who play all three sports (C&F&H) has a number greater than zero. If the five disjoint areas described *do not include* the intersection of all three sets (C&F&H), then the number of players in that region is zero by implication. In this scenario, statement C would be false (“There are NO hockey players who play both cricket and football”).
Let’s test if this interpretation makes sense by assuming statement C is indeed the “not correct” one. This means C&F&H = 0. The five regions could then be, for example: C only, F only, H only, C&F only, C&H only, F&H only (this is 6, so one more must be 0). Or perhaps C only, F only, H only, C&F only, F&H only (omitting C&H only and C&F&H). Or C only, F only, H only, C&F only, C&H only (omitting F&H only and C&F&H).
If C&F&H = 0, then statement C is false. Let’s see if the other statements *could* be true under various configurations of 5 non-empty regions (where C&F&H=0):
A) “Every hockey player plays football”: Requires H only = 0 and C&H only = 0. Possible if the 5 regions are, for example, F only, C only, C&F only, F&H only, some other.
B) “Every cricket player plays either football or hockey”: Requires C only = 0. Possible.
D) “There are some football players who play neither cricket nor hockey”: Requires F only > 0. Possible.
Since statement C (C&F&H > 0) directly contradicts the implication that the region C&F&H is one of the *missing* regions among the five non-empty ones, statement C is the most likely to be the “not correct” one under a plausible interpretation of the five given areas. The wording “five disjoint bounded areas… labelled with number of players belonging to that area” strongly suggests these are the *only* regions within the shapes that contain players. If the C&F&H intersection is not one of these five, its count is 0.