In a class of 33, 20 play cricket, 25 play football and 18 play hockey. 15 play both cricket and...

1. Translate the problem requirements: We need to find students who play ONLY cricket (cricket but neither football nor hockey). Given information includes total students (33), individual sport counts, pairwise overlaps, and that everyone plays at least one sport.
2. Find students playing all three sports: Use the principle that total students equals the sum of individual groups minus overlaps plus those in all three groups.
3. Determine cricket-only players using subset analysis: Break down the 20 cricket players into those who also play other sports versus those who play cricket exclusively.
4. Verify the answer through logical consistency: Check that our answer makes sense given the constraints and relationships in the problem.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find.

We have 33 students total, and each student plays at least one sport. Think of this like a survey where everyone must check at least one box.

Given information:

• 20 students play cricket
• 25 students play football
• 18 students play hockey
• 15 students play both cricket AND football
• 12 students play both football AND hockey
• 10 students play both cricket AND hockey

What we need to find: Students who play ONLY cricket (meaning they play cricket but neither football nor hockey).

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Find students playing all three sports

To solve this systematically, we need to figure out how many students play all three sports.

Here's the key insight: When we add up all the individual sport totals (\(20 + 25 + 18 = 63\)), we're counting some students multiple times. We need to subtract the overlaps, but then we've subtracted the "all three" group too many times.

Using the inclusion-exclusion principle in plain English:
Total = (Individual totals) - (Pairwise overlaps) + (All three overlaps)

\(33 = (20 + 25 + 18) - (15 + 12 + 10) + \mathrm{(All\ three)}\)
\(33 = 63 - 37 + \mathrm{(All\ three)}\)
\(33 = 26 + \mathrm{(All\ three)}\)
\(\mathrm{All\ three} = 33 - 26 = 7\)

So 7 students play all three sports.

Process Skill: INFER - Drawing the non-obvious conclusion about the three-way overlap

3. Determine cricket-only players using subset analysis

Now let's break down the 20 cricket players into different categories:

Think of the cricket players as being in different "zones":

• Zone 1: Play cricket AND football AND hockey = 7 students (from step 2)
• Zone 2: Play cricket AND football BUT NOT hockey = \(15 - 7 = 8\) students
• Zone 3: Play cricket AND hockey BUT NOT football = \(10 - 7 = 3\) students
• Zone 4: Play ONLY cricket = ?

The total cricket players must equal the sum of all these zones:
\(20 = 7 + 8 + 3 + \mathrm{(cricket\ only)}\)
\(20 = 18 + \mathrm{(cricket\ only)}\)
\(\mathrm{Cricket\ only} = 20 - 18 = 2\)

So 2 students play only cricket.

Process Skill: VISUALIZE - Breaking down overlapping groups into distinct categories

4. Verify the answer through logical consistency

Let's double-check our answer makes sense:

• We found 2 students play only cricket
• This seems reasonable given that cricket has the smallest individual count (20)
• Most cricket players also play other sports, which fits with the high overlap numbers
• Our answer of 2 matches choice D

We can also verify by checking if all our numbers add up correctly across all sports, and they do.

Final Answer

The number of students who play only cricket is 2.

Answer: D

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "both" as "exactly two" instead of "at least two"

Students often misread "15 play both cricket and football" as meaning these students play exactly these two sports and no others. This leads them to incorrectly assume these 15 students don't play hockey, when in reality some could play all three sports. This misinterpretation makes it impossible to use the inclusion-exclusion principle correctly.

2. Attempting to solve without finding the three-way overlap first

Many students try to jump directly to finding "cricket only" players without recognizing that they need to determine how many play all three sports first. They might try to work backwards from individual sport totals, leading to incorrect categorization of players and wrong final answers.

3. Confusing the constraint "each student plays at least one game"

Some students misinterpret this constraint as meaning each student plays exactly one game, or they ignore it entirely. This fundamental misunderstanding affects their entire approach, as they fail to recognize this is a complete coverage problem where the total must equal 33.

Errors while executing the approach

1. Arithmetic errors in the inclusion-exclusion calculation

When calculating \(33 = (20 + 25 + 18) - (15 + 12 + 10) + \mathrm{(All\ three)}\), students frequently make simple addition or subtraction mistakes. Common errors include calculating \(20 + 25 + 18 = 73\) instead of 63, or \(15 + 12 + 10 = 47\) instead of 37, leading to an incorrect value for students playing all three sports.

2. Incorrectly calculating exclusive pairwise overlaps

When finding "cricket AND football BUT NOT hockey," students often forget to subtract the all-three group and use 15 directly instead of \(15 - 7 = 8\). This error propagates through their zone analysis, giving them wrong counts for each category of cricket players.

Errors while selecting the answer

1. Selecting an intermediate calculation instead of the final answer

Students might correctly calculate that 7 play all three sports or 8 play cricket and football but not hockey, then mistakenly select one of these intermediate values as their final answer instead of the 2 students who play only cricket. They lose track of what the question is actually asking for.

2. Double-checking against the wrong constraint

When verifying their answer, students might check whether their cricket-only count seems reasonable compared to football or hockey totals, rather than checking if all their zone calculations add up to the given sport totals. This leads them to second-guess a correct answer of 2 because it seems "too small" compared to other sports.