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In a class of 33, 20 play cricket, 25 play football and 18 play hockey. 15 play both cricket and football, 12 play football and hockey and 10 play cricket and hockey. If each student plays at least one game, how many students play only cricket?
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:
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
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
Now let's break down the 20 cricket players into different categories:
Think of the cricket players as being in different "zones":
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
Let's double-check our answer makes sense:
We can also verify by checking if all our numbers add up correctly across all sports, and they do.
The number of students who play only cricket is 2.
Answer: D
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.
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.
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.
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.
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.
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.
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.