In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey...

1. Translate the problem requirements: We need to find how many students play exactly two sports. This means students who play Hockey and Cricket but not Football, OR Cricket and Football but not Hockey, OR Hockey and Football but not Cricket.
2. Set up the inclusion-exclusion framework: Use the given information to establish relationships between total students, those playing sports, and overlaps between different sports.
3. Find students playing all three sports: Use the total number of students playing at least one sport (\(50 - 18 = 32\)) and apply inclusion-exclusion to determine how many play all three sports.
4. Calculate exactly two sports: For each pair of sports, subtract those who play all three from those who play both sports to get the count of students playing exactly that pair.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're looking for in everyday terms. When the problem asks for students who play "exactly two sports," we mean students who play two sports but NOT the third one.

Think of it this way:

• Some students play Hockey and Cricket, but we only want to count those who DON'T also play Football
• Some students play Cricket and Football, but we only want to count those who DON'T also play Hockey
• Some students play Hockey and Football, but we only want to count those who DON'T also play Cricket

So we need to subtract the "triple players" (those who play all three sports) from each pair to get our "exactly two" counts.

Process Skill: TRANSLATE - Converting the phrase "exactly two sports" into a clear mathematical requirement

2. Set up the inclusion-exclusion framework

Let's organize our information in plain English first:

• Total students: 50
• Students not playing any sport: 18
• Therefore, students playing at least one sport: \(50 - 18 = 32\)

Now for the individual sports and overlaps:

• Hockey players: 20
• Cricket players: 15
• Football players: 11
• Hockey AND Cricket players: 7
• Cricket AND Football players: 4
• Hockey AND Football players: 5
• Hockey AND Cricket AND Football players: Let's call this x (unknown)

The inclusion-exclusion principle tells us that when we add up all the individual sport players, we double-count the pairs, and triple-count those who play all three.

3. Find students playing all three sports

Using inclusion-exclusion in everyday language: If we add all individual sport players (\(20 + 15 + 11 = 46\)), we've counted each "pair player" twice and each "triple player" three times.

To get the actual count of 32 students playing at least one sport, we need to:
\(46 - \text{(pairs we over-counted)} + \text{(triples we under-counted after the correction)} = 32\)

The pairs we over-counted: \(7 + 4 + 5 = 16\)
So: \(46 - 16 + \mathrm{x} = 32\)
\(30 + \mathrm{x} = 32\)
Therefore: \(\mathrm{x} = 2\)

So 2 students play all three sports.

Process Skill: INFER - Recognizing that we need to work backwards from the total to find the missing piece

4. Calculate exactly two sports

Now we can find students playing exactly two sports by subtracting the "all three" players from each pair:

Students playing exactly Hockey and Cricket (but not Football):
\(7 - 2 = 5\) students

Students playing exactly Cricket and Football (but not Hockey):
\(4 - 2 = 2\) students

Students playing exactly Hockey and Football (but not Cricket):
\(5 - 2 = 3\) students

Total students playing exactly two sports:
\(5 + 2 + 3 = 10\) students

4. Final Answer

The number of students who play exactly two of these sports is 10.

This matches answer choice B.

Verification: Let's check our work by counting all categories:

• Only Hockey: \(20 - 5 - 3 - 2 = 10\)
• Only Cricket: \(15 - 5 - 2 - 2 = 6\)
• Only Football: \(11 - 2 - 3 - 2 = 4\)
• Exactly two sports: 10 (our answer)
• All three sports: 2
• No sports: 18

Total: \(10 + 6 + 4 + 10 + 2 + 18 = 50\) ✓

Common Faltering Points

Errors while devising the approach

• Misinterpreting "exactly two sports": Students often confuse "exactly two sports" with "at least two sports." They might directly use the given overlap numbers (7 for Hockey and Cricket, 4 for Cricket and Football, 5 for Hockey and Football) without realizing these represent students who play AT LEAST those two sports, which could include students playing all three.
• Not recognizing the need for inclusion-exclusion: Students may attempt to solve this as a simple addition/subtraction problem without understanding that they need to use the inclusion-exclusion principle to find the number of students playing all three sports first.
• Overlooking the constraint about non-players: Students might ignore the critical information that 18 students don't play any sport, which means only 32 students play at least one sport. This constraint is essential for finding the unknown number of students playing all three sports.

Errors while executing the approach

• Arithmetic errors in inclusion-exclusion formula: Students may make calculation mistakes when applying the inclusion-exclusion principle: \(|\mathrm{H} \cup \mathrm{C} \cup \mathrm{F}| = |\mathrm{H}| + |\mathrm{C}| + |\mathrm{F}| - |\mathrm{H} \cap \mathrm{C}| - |\mathrm{C} \cap \mathrm{F}| - |\mathrm{H} \cap \mathrm{F}| + |\mathrm{H} \cap \mathrm{C} \cap \mathrm{F}|\), particularly when substituting values like \(32 = 20 + 15 + 11 - 7 - 4 - 5 + \mathrm{x}\).
• Incorrectly calculating "exactly two" categories: After finding that 2 students play all three sports, students may forget to subtract this number from each pairwise intersection, or they may subtract it incorrectly from the wrong values.

Errors while selecting the answer

• Adding the wrong values for final answer: Students may correctly calculate the individual "exactly two" categories (5 for Hockey-Cricket only, 2 for Cricket-Football only, 3 for Hockey-Football only) but then make an error when adding them together, or they might accidentally include the "all three sports" count in their final sum.