Question

In the following figure, the Pentagon, the Rectangle, the Triangle and the Circle represent the number of students in a class who play Cricket, Soccer, Hockey and Baseball respectively. Which of the following is the highest? 

• A. Number of students who play at least one of Hockey or Baseball.
• B. Number of students who play exactly two of the sports.
• C. Number of students who play exactly one of the four sports.
• D. Number of students who play at least one of Cricket or Soccer.

Hint:

On union/comparison questions with region-labelled Venn/Euler diagrams, first spot the largest disjoint “only” regions that a choice includes. A union that already packs multiple big disjoint blocks will almost always beat counts restricted to “exactly one/two” or to a smaller union.

Answer 1

The Correct Option is D

Step 1: Decode the sets.
Let \(C\) = Cricket (pentagon), \(S\) = Soccer (rectangle), \(H\) = Hockey (triangle), \(B\) = Baseball (circle). The numbers written in each region of the diagram are \emph{counts of students} in that exact region.
Step 2: Read the big “only” regions.
From the diagram, two conspicuously large regions are:
\(\bullet\) \(C\)–only has \(\,10\,\) students (top of the pentagon).
\(\bullet\) \(S\)–only has \(\,9\,\) students (right part of the rectangle).
These two parts are \emph{disjoint} and both are included in \(C \cup S\). Hence, \[ |C \cup S| \;\ge\; 10 + 9 \;=\; 19, \] \emph{before} even counting any overlaps involving \(C\) or \(S\) with the other sports. Therefore \(C \cup S\) is already quite large.
Step 3: Compare with Hockey or Baseball.
For option (a) we need \(|H \cup B|\). Its “only” regions are not both as large as \(10\) and \(9\) simultaneously; the largest single “only” area among \(H\) and \(B\) is \(9\) (Baseball–only at the bottom circle). Even after adding the other smaller \(H\)-only portion and shared regions, \(|H \cup B|\) cannot outgrow \(|C \cup S|\), because all overlaps that include \(C\) or \(S\) are also pulling \(|C \cup S|\) further \emph{above} its baseline of \(19\). Thus, \[ |H \cup B| \;<\; |C \cup S|. \] Step 4: Compare with “exactly one” and “exactly two”.
Options (b) and (c) ask for totals of “exactly two” and “exactly one” sports, which are formed by \emph{selected} regions only. Such totals exclude many regions (e.g., all triples/quadruple overlaps or all non-qualifying areas) and are therefore necessarily smaller than a large union like \(|C \cup S|\) that includes every region touching \(C\) or \(S\) (including all overlaps). Hence both (b) and (c) are less than \(|C \cup S|\).
Step 5: Conclusion.
Because the union \(C \cup S\) contains two large disjoint “only” blocks \(10\) and \(9\) \emph{plus} all overlaps that touch \(C\) or \(S\), it dominates the other counted quantities. Therefore the highest among the four given options is (d) \(|C \cup S|\).
\(\boxed{\text{Number of students who play at least one of Cricket or Soccer is the highest.}}\)