最大两门课程重叠

Created on: December 20, 2024

Answered using O1 by Chat01

Question

一个班级总共有30名学生。心理学、社会学和市场营销专业的学生人数分别是总人数的 $\frac{2}{5}$ 、 $\frac{4}{15}$ 和 $\frac{1}{3}$ 。问：这三门课程中，最多有多少名学生同时报名了其中的两门课程？

Thought

Identifying student distribution

I calculated the student distribution across Psychology, Sociology, and Marketing courses, resulting in class sizes of 12, 8, and 10, respectively.

Maximizing set intersections

I’m calculating the maximum number of students enrolled in two or more courses by maximizing overlaps while adhering to set size limits and ensuring the sum of enrollments equals the total number of students.

Piecing together constraints

Using the principle of inclusion-exclusion, I account for all students enrolled in Psychology, Sociology, and Marketing, ensuring the sum of enrollments equals the total number of students.

Ensuring accurate counts

To accurately determine the number of students taking at least two courses, I adjust for students counted in all three sets to avoid overestimation.

Adjusting intersection counts

I'm working through the formula to accurately count students enrolled in at least two courses, ensuring the sum of pairwise intersections equals the triple intersection.

Evaluating student course overlap

Taking a closer look at student enrollments in Psychology, Sociology, and Marketing shows that with 30 students no overlaps are needed. This maximizes the number of students taking exactly two courses.

Maximizing student enrollments

Ensuring maximum student enrollments in Psychology, Sociology, and Marketing involves placing students first in all three sets, targeting the smallest set size to maximize intersection.

Calculating student spread

I worked out student enrollments across Psychology, Sociology, and Marketing, ensuring the sum of enrollments matches the total class size while considering all potential overlaps and individual enrollments.

Clarifying course enrollment

I'm digging into the interpretation of "两门课程" to determine if it means exactly two or at least two enrollments, which significantly impacts the solution's accuracy.

Clarifying enrollments

I'm working through the ambiguous phrasing to determine if "两门课程" means exactly two enrollments or if students might be taking at least two courses simultaneously. This interpretation is key to solving the problem.

Clarifying student choices

I’m thinking through the interpretation of enrollment fractions, ensuring all 30 students are accounted for without exceeding the total class size. This involves clarifying the meaning of "specialties" or "classes chosen."

Evaluating enrollment patterns

I’m piecing together enrollment fractions to ensure each student is counted once. This setup implies no overlap among students, which might not align with the question's expectations.

Verifying intersections

Ensuring the sum of pairwise intersections equals the triple intersection is crucial for accurate student enrollment distribution across courses. This confirms the total accounts for all class members.

Breaking down intersections

I’m examining pairwise and triple intersections, ensuring the sums align with set sizes to accurately map out student enrollments across Psychology, Sociology, and Marketing.

Revising enrollment calculations

Thinking through each set by defining unique enrollments and ensuring intersections align with the total class size, I’m moving towards more accurate enrollment counts.

A new assumption

Considering the possibility that not every student enrolled in a course, which may resolve the inconsistency in intersection assumptions. This leads me to a clearer path forward.

Maximizing enrollments

I’m focusing on reducing the number of students taking no courses and those enrolled solely in one course to boost the count of students enrolled in at least two courses.

Reassessing student counts

I'm re-evaluating student enrollments across Psychology, Sociology, and Marketing, acknowledging that not everyone might be taking these courses and emphasizing the need for a balanced approach.

Clarifying course interpretations

I’m thinking through the ambiguity in the phrase "同时报名两门课程" and interpreting it as enrolling in two courses simultaneously, which might include three courses.

Navigating interpretation

I’m gathering info on the ambiguity of the phrase "两门课程" to determine if it means exactly two or at least two enrollments, which shapes the solution approach.

Maximizing student enrollments

I’m focusing on maximizing students enrolled in exactly two courses, likely considering the more straightforward scenario of two courses instead of grappling with the ambiguity of at least two courses.

Maximizing enrollments

I’m focusing on maximizing students enrolled in exactly two courses by setting up equations for pairwise intersections from Psychology, Sociology, and Marketing, and solving for variables to achieve this.

Maximizing enrollments

I’ve determined ps=5, pm=7, sm=3, resulting in 15 students in exactly two courses. By setting single enrollments and no course students to zero, I’m focusing on maximizing overlapping enrollments.

Revisiting course requirements

I’m noting the academic requirement for all students to enroll in at least one course, which suggests zero students should be without course selections.

Reassessing student choices

I’m evaluating course selections by excluding single enrollments and ensuring all students are choosing at least one course for a more realistic distribution.

Ensuring complete participation

I’ve confirmed that no triple intersection occurs when every student is enrolled in at least one course, leading to the final conclusion that maximizing enrollments hinges on excluding single enrollments.

Breaking down enrollment

I’m simplifying enrollment patterns by excluding single course students, ensuring all participants are in at least one course, and refining intersections for a more accurate assessment.

Ensuring enrollment consistency

I’m working on confirming that the enrollment data aligns with the class's total, recognizing the possibility that some students might not be enrolled in any of the three courses.

Redistributing enrollments

I’m mapping out the distribution of enrollments across Psychology, Sociology, and Marketing, ensuring the total includes all students while adjusting for the possibility of some not participating in any course.

Revisiting enrollments

I’m working through the distribution of students in Psychology, Sociology, and Marketing, aiming to ensure all students are accounted for and adjusting for any not attending any course.

Maximizing enrollments

I’m thinking through how to maximize students enrolled in at least two courses by increasing the number of students without a course and minimizing the triple intersection.

Maximizing overlaps

I’m working to maximize students in at least two courses by increasing students without a course and minimizing the triple intersection, while ensuring the set sizes align with given proportions.

Mapping out intersections

I’m thinking through optimal student enrollments in courses by setting d=8 to maximize triple intersection, ensuring the total class size and student enrollment proportions are maintained.

Maximizing student enrollments

I’m exploring maximizing double intersections by setting d=0 and focusing on pairwise intersections, ensuring all students are accounted for while maintaining set size constraints.

Optimizing course enrollments

I’m refining student enrollments by maximizing double intersections with set values, ensuring all are enrolled, and confirming totals align with class size constraints. The goal is to enhance accuracy.

Balancing intersections

I’m thinking about maximizing double intersections by carefully balancing enrollments between S and P, ensuring M's needs are met without overloading any individual set. This refining process guides enrollment strategy.

Balancing student enrollments

I’m confirming a balanced approach, ensuring no course is overcrowded. Adjusting enrollments maintains the class size and set distribution while maximizing the number of students in at least two courses.

Simplifying student distribution

I'm focusing on calculating student enrollments by streamlining the equation system for pairwise intersections, ensuring all students are enrolled at least twice, and verifying the class size constraints.

Ensuring enrollments

I’ve confirmed that all 30 students are accounted for, with 15 distinct students each enrolled in exactly two courses, effectively balancing the student enrollment across the required courses.

Maximizing student enrollments

I'm piecing together how adding a triple intersection might reduce the count of students in exactly two courses, aiming to maintain a balanced student distribution across all courses.

Testing combinations

I’m thinking through different triple intersections and singles to fine-tune the student enrollments, aiming to maximize the number of students enrolled in exactly two courses while keeping the total class size accurate.

Maximizing enrollments

I'm thinking that the optimal enrollment scenario involves 15 students taking exactly two courses, with no single or triple intersections, balancing the need for distinct enrollments across Psychology, Sociology, and Marketing.

Ensuring enrollment precision

I’m thinking through the optimal student enrollment in Psychology, Sociology, and Marketing, ensuring the total class size and specific course enrollments are accurately balanced.

Maximizing enrollments

I’m thinking about adding a triple intersection, but it reduces the 15 students in exactly two courses and uses up 3 seats, making it impossible to reach the required course enrollments.

Thought about课程重叠问题，持续 2m 35s

Answer

分析过程（详细推导）：

设这30名学生中选择心理学课程的有：
$|P| = \frac{2}{5} \times 30 = 12$ 人

选择社会学课程的有：
$|S| = \frac{4}{15} \times 30 = 8$ 人

选择市场营销课程的有：
$|M| = \frac{1}{3} \times 30 = 10$ 人

因此，三个集合的基数为：
$|P|=12, \quad |S|=8, \quad |M|=10.$

总计有30名学生，但题目并未明确说明所有学生都至少选了一门课程，因此允许一部分学生一门课也没选。

我们用集合运算中的标准符号：

设选择仅心理学的为 $x$ 、仅社会学的为 $y$ 、仅市场营销的为 $z$ 。
设选择正好两门课程的学生数分别为：
- $ps$ 表示选了心理学和社会学但没选市场营销的学生数
- $pm$ 表示选了心理学和市场营销但没选社会学的学生数
- $sm$ 表示选了社会学和市场营销但没选心理学的学生数
设选择三门课程的学生数为 $d$ 。
没有选任一课程的学生数为 $w$ 。

因此总人数为：
$x+y+z+ps+pm+sm+d+w = 30.$

每门课的人数要求为：

对心理学（P）：
$x + ps + pm + d = 12.$
对社会学（S）：
$y + ps + sm + d = 8.$
对市场营销（M）：
$z + pm + sm + d = 10.$

我们希望最大化“同时报名了其中的两门课程”的学生数量。
这里“同时报名了两门课程”一般理解为“至少两门课”。（若理解为恰好两门亦可，后续会发现相同构造可达最大值）。
"至少两门"的学生包括正好两门(ps+pm+sm)和三门(d)，即：
$\text{至少两门} = ps + pm + sm + d$

为了尽可能增大此数量，我们尝试让更多学生参加多门课程，并尽可能减少只选一门或不选课的人数。但需要注意各科人数的限制。

尝试构造最大两门以上重叠的方案：

观察到三个课程人数总和：
$|P| + |S| + |M| = 12 + 8 + 10 = 30.$

如果所有参加课程的学生恰好各选两门课，那么每位学生贡献“2个选课人数”。要达到总共30个选课计数（12+8+10=30），如果每个人都是两门课，则需要 $\frac{30}{2} = 15$ 名学生（因为每人贡献2个选课计数，15人两门课合计提供30个选课人数刚好满足三个课程的总人数要求）。

这意味着，如果能构造出 exactly 15个学生，每人选两门课（无单门无三门），就能完美满足各科人数要求，而且这15人都属于“至少两门课”这一类，数量为15。而其余(30-15)=15名学生则一门不选即可。

验证是否可行：

要求：

$|P|=12$
$|S|=8$
$|M|=10$

尝试用两两交集分配：
我们有三种两两组合： $ps, pm, sm$ 。

需要满足：
$ps + pm = |P|=12 - d - x$ 若无单门无三门，且不考虑单门，则 $d=0, x=0$ 。此时：
$ps + pm = 12$

$ps + sm = 8$

$pm + sm = 10$

无单门无三门(d=0,x=0,y=0,z=0)，只解这三个方程：

从前两式中可解出：
(1) $ps + pm = 12$
(2) $ps + sm = 8$

由(1)得 $pm = 12 - ps$
由(2)得 $sm = 8 - ps$

再利用 $pm + sm =10$ ：
$(12 - ps) + (8 - ps) = 10$
$20 - 2ps =10$
$2ps =10$
$ps=5.$

代回：
$pm =12 -5=7$
$sm =8 -5=3.$

检查：
$pm+sm=7+3=10$ 正好满足。

此时：
$ps=5, pm=7, sm=3, d=0, x=y=z=0$

再验证各课程人数：

P课：ps(5人)+pm(7人)=12人 ✅
S课：ps(5人)+sm(3人)=8人 ✅
M课：pm(7人)+sm(3人)=10人 ✅

完美满足各课人数要求。

此方案共有
$ps+pm+sm=5+7+3=15$ 名学生，每人选两门课，没有三门、没有单门课学生。这15人恰好构成所有选课的学生，其余15人一门不选(w=15)。

这样一来，"同时报名了两门课程"的学生数就是15人。这显然是最大值，因为这样每位上课的学生都贡献了2个课程选择，最大化了两门以上选课人数。

结论：
在满足给定人数比例的前提下，最多有15名学生能同时选修两门课程。

最终答案：
$\boxed{15}$

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