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27 Nov 2020, 07:10
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None: 60
All three: 10
Out of the 150 students at Rocket Brown Elementary School, 90 joined the football team, 50 joined the tennis team, 40 joined the hockey team, and 70 joined exactly two of these teams.
Select for None the maximum number of students who could have chosen not to join any of the three teams, and select for All three the maximum number of students who could have joined all three teams.
Select for None the maximum number of students who could have chosen not to join any of the three teams, and select for All three the maximum number of students who could have joined all three teams.
| None | All three | |
| 0 | ||
| 10 | ||
| 20 | ||
| 40 | ||
| 50 | ||
| 60 |
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Official Answer
None: 60
All three: 10
27 Nov 2020, 07:10
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Official Solution:
Check the diagram below:
The formula for three overlapping sets is:
\(Total = A + B + C - (sum \ of \ overlaps \ between \ exactly \ 2 \ groups) - 2 \times (all \ three) + None\).
This formula adjusts for overlaps when adding A, B, and C. Sections representing those in exactly two groups (d, e, and f) are counted twice, so we subtract them once to ensure they are only counted once. Similarly, the section representing those in all three groups (g) is counted three times, so we subtract it twice to count it only once.
For the original question, substituting the values gives:
\(150 = 90 + 50 + 40 - 70 - 2 \times (all \ three) + None\).
\(2 \times (all \ three) + 40 = None\).
Notice that the above implies maximizing All three also maximizes None, and vice versa. Essentially, when one is maximized, both are maximized together.
Further notice that the maximum number of students who could have chosen not to join any of the three teams cannot exceed 60, as we know that out of 150 students, 90 joined the football team. But can the maximum for None actually be 60? For this to happen, all students in the hockey and tennis teams would need to also be part of the football team, ensuring that no additional students are outside the football team.
If the maximum for None is 60, using the equation \(2 \times (all \ three) + 40 = None\), it follows that All three would be 10. This gives the following scenario:
Thus, this is a perfectly valid case. Since we concluded that maximizing None also maximizes All three, the maximum for All three is 10.
Correct answer:
None "60"
All three "10"
TetyanaRokobit
Check the diagram below:
The formula for three overlapping sets is:
\(Total = A + B + C - (sum \ of \ overlaps \ between \ exactly \ 2 \ groups) - 2 \times (all \ three) + None\).
This formula adjusts for overlaps when adding A, B, and C. Sections representing those in exactly two groups (d, e, and f) are counted twice, so we subtract them once to ensure they are only counted once. Similarly, the section representing those in all three groups (g) is counted three times, so we subtract it twice to count it only once.
For the original question, substituting the values gives:
\(150 = 90 + 50 + 40 - 70 - 2 \times (all \ three) + None\).
\(2 \times (all \ three) + 40 = None\).
Notice that the above implies maximizing All three also maximizes None, and vice versa. Essentially, when one is maximized, both are maximized together.
Further notice that the maximum number of students who could have chosen not to join any of the three teams cannot exceed 60, as we know that out of 150 students, 90 joined the football team. But can the maximum for None actually be 60? For this to happen, all students in the hockey and tennis teams would need to also be part of the football team, ensuring that no additional students are outside the football team.
If the maximum for None is 60, using the equation \(2 \times (all \ three) + 40 = None\), it follows that All three would be 10. This gives the following scenario:
Thus, this is a perfectly valid case. Since we concluded that maximizing None also maximizes All three, the maximum for All three is 10.
Correct answer:
None "60"
All three "10"
03 Dec 2024, 01:42
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I have revised the question, solution, and formatting by adding more details to enhance clarity.
