We can approach this problem using set theory and Venn diagrams to systematically count the number of students majoring in various combinations of Chemistry (C), Physics (P), and Mathematics (M).

Let's define the following:

• $|C| = 10$ is the number of students who major in Chemistry.
• $|P| = 13$ is the number of students who major in Physics.
• $|M| = ?$ is the number of students who major in Mathematics (this is what we need to find in part (a)).
• $|C \cap P| = 7$ is the number of students who major in both Chemistry and Physics.
• $|C \cap M| = 6$ is the number of students who major in both Chemistry and Mathematics.
• $|P \cap M| = 8$ is the number of students who major in both Physics and Mathematics.
• $|C \cap P \cap M| = 5$ is the number of students who major in all three subjects.

Part (a): How many students major in Mathematics?

We want to find the total number of students majoring in Mathematics, $|M|$.

We can use the inclusion-exclusion principle for three sets (Chemistry, Physics, and Mathematics) to calculate the total number of students who major in Mathematics. The inclusion-exclusion principle states:

[ |M| = |M \cap C^c \cap P^c| + |M \cap C \cap P^c| + |M \cap P \cap C^c| + |M \cap C \cap P| + |M \cap P \cap C| - |M \cap