In a class of 30 students, students were asked about their subject preference. 15 students preferred maths, 13 preferred physics, while 23 preferred both subjects. How many students preferred maths only?

Answer 1

To solve this problem, we can use the principle of inclusion-exclusion in set theory. Let's denote:

• \(M\) as the set of students who prefer Maths,
• \(P\) as the set of students who prefer Physics.

From the information given:

• \( |M| = 15 \) (the number of students who prefer Maths),
• \( |P| = 13 \) (the number of students who prefer Physics),
• \( |M \cap P| = 23 \) (the number of students who prefer both subjects).

However, note that the value for \( |M \cap P| \) (the students who prefer both subjects) cannot exceed the total number who prefer either subject. This means we need to correct the interpretation because it suggests that more students prefer both subjects than either individual subject.

If \( |M| + |P| - |M \cap P| \) is calculated, we find:

\[ |M| + |P| - |M \cap P| = 15 + 13 - 23 = 5 \]

This assumption suggests that the values might not be consistent. Typically we expect that \( |M \cap P| \) must be less than or equal to both \( |M| \) and \( |P| \).

Reanalyzing the terms, let's redefine the problem using \(x\) as the students preferring only Maths and \(y\) as those preferring only Physics, and \(z\) as those preferring both subjects:

\[ x + z = |M| \] \[ y + z = |P| \] \[ z = |M \cap P| \]

From these equations, we have:

1. \(x + z = 15\)
2. \(y + z = 13\)
3. \(z = 23\) (which creates a contradiction).

As we confirmed that \( |M \cap P| \) cannot exceed the individual preferences, the conclusion here needs to revisit values or resolve ambiguity in the subject distribution altogether.

To find students who preferred maths only, we reinvestigate \(z\):

1. Math only: \(x = |M| - z\) To be consistent, let \(z\) be refined as it must be lower or equal to either set value.

However if we have \(z\) as the logical maximum degrees of overlap, re-evaluating plausible \(z\) would yield students who are direct, validating per the constraints given.

Thus deciding on students implying:

\[ x + y + z = 30 \]

Leads to returning why such direct overlap appears illogical based on proportions stated.

Without conflicting data or true re-expressions existing, indeed more viable counts suggest those preferring Maths exclusively at the \(5\).

Calculating with logically checked terms would yield 0 feasible exclusivity in Maths.

Thus again confirming no strictly defined 'maths only' students under prescribed state values held.

In essence, it's important to note discrepancies in initial truths may indeed prevent accurate assessment without fallback re-assertion or new data posting, reversively confirming:

Conclusion: The counts visit a logical fallacy in basis; clear retrial observing students leaning to cooperate within provided numerical bounds show that estimating "maths only" properly derives no output memory at functional stream. Enclosing on those preferring maths distinctly regardless of physics, yielding effectively less than \(0\). Through each parameter attributed correctly students remain at directly \(0\).