There are 20 students in a science class. All are science majors each majors in Chemistry, Physics, or Mathematics, including some who major in two or even three of those subjects. Of those 20 students:

10 major in Chemistry (and, possibly, in other subjects),

13 major in Physics (and, possibly, in other subjects),

7 major in both Chemistry and Physics (and, possibly, in Math),

6 major in both Chemistry and Mathematics (and, possibly, in Physics),

8 major in both Mathematics and Physics (and, possibly, in Chemistry),

5 major in all the three subjects.

(a) How many students major in Mathematics?

(b) If a random Mathematics major is selected, what is the probability that this student majors in exactly two subjects?

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

There are 20 students in a science class. All are science majors each majors in Chemistry, Physics, or Mathematics, including some who major in two or even three of those subjects. Of those 20 students: 10 major in Chemistry (and, possibly, in other subjects), 13 major in Physics (and, possibly, in other subjects), 7 major in both Chemistry and Physics (and, possibly, in Math), 6 major in both Chemistry and Mathematics (and, possibly, in Physics), 8 major in both Mathematics and Physics (and, possibly, in Chemistry), 5 major in all the three subjects. (a) How many students major in Mathematics? (b) If a random Mathematics major is selected, what is the probability that this student majors in exactly two subjects?

Learn how to solve a complex problem involving 20 science students majoring in Chemistry, Physics, and Mathematics using set theory and the inclusion-exclusion principle. Find how many students major in Mathematics and the probability that a randomly selected math major studies exactly two subjects. Suitable for students in Grades 10-12.

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