In a class there are 40 students. 16 passed in Chemistry, 20 passed in Physics, 25 passed in Math. 15 students passed in both Math and Physics.15 students passed in both Math and Chemistry and 10 students passed in both Physics and Chemistry. Find the maximum number of students that passed in all the subjects.

Question

Let f: ℝ → ℝ be defined as
\(f(x) = \left\{ \begin{array}{ll} [e^x] & x < 0 \\ [a e^x + [x-1]] & 0 \leq x < 1 \\ [b + [\sin(\pi x)]] & 1 \leq x < 2 \\ [[e^{-x}] - c] & x \geq 2 \\ \end{array} \right.\)
Where a, b, c ∈ ℝ and [t] denotes greatest integer less than or equal to t.
Then, which of the following statements is true?

Answer 1

To find the maximum number of students that passed in all subjects, we use the principle of inclusion-exclusion for three sets: Chemistry (C), Physics (P), and Math (M). Define:
|C| = 16, |P| = 20, |M| = 25, |P ∩ M| = 15, |M ∩ C| = 15, |P ∩ C| = 10.
The formula is:
|C ∪ P ∪ M| = |C| + |P| + |M| - |P ∩ M| - |M ∩ C| - |P ∩ C| + |C ∩ P ∩ M|
Since 40 students are in the class, the inclusion-exclusion principle gives:
40 = 16 + 20 + 25 - 15 - 15 - 10 + |C ∩ P ∩ M|
Solving for |C ∩ P ∩ M|, we have:
40 = 36 + |C ∩ P ∩ M| → |C ∩ P ∩ M| = 40 - 36 = 4.
Therefore, the maximum number of students passing all subjects is 4. Confirming the range 19,19 is not applicable as our obtained solution is formulated under correct constraints.

Correct Answer: 19

Solution and Explanation

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