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### Question 1:

We have 20 students, with the following information provided:
- 10 major in Chemistry
- 13 major in Physics
- 13 major in Mathematics
- 7 major in both Chemistry and Physics
- 6 major in both Chemistry and Mathematics
- 8 major in both Mathematics and Physics
- 5 major in all three subjects (Chemistry, Physics, Mathematics)

We are asked:
(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?

#### Solution:

**(a) How many students major in Mathematics?**

Using the principle of inclusion-exclusion, we can find how many students major in Mathematics. Here's the breakdown:

- Total students majoring in Mathematics: 13.
- Subtract those who major in both Chemistry and Mathematics: 6.
- Subtract those who major in both Physics and Mathematics: 8.
- Add back the students who major in all three subjects (since they were subtracted twice): 5.

Now, let's calculate step by step to determine the exact number of students who major only in Mathematics. We will exclude those who major in two or three subjects.

**(b) The probability of selecting a Mathematics major who also majors in exactly two subjects.**

### Question 2:

There are six people forming three couples (A-B, C-D, E-F) sitting on a long bench, with all seating configurations equally likely.

(a) What is the total number of possible seating configurations where people from each couple sit next to each other?  
(b) What is the probability that B ends up sitting next to C, assuming couples sit together?

#### Solution:

**(a) Total number of seating configurations:**

The couples must sit together, so treat each couple as a single unit. Since there are three couples, the number of ways to arrange them is \( 3! \) (since they can be rearranged in 3 factorial ways). However, within each couple, the two members can switch places, so for each couple, there are \( 2! \) ways. The total number of seating configurations is \( 3! \times 2! \times 2! \times 2! \).

**(b) Probability that B ends up sitting next to C:**

Since we're only considering seating arrangements where couples sit together, we calculate the probability of B sitting next to C within these constraints.

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Would you like me to solve this in more detail or clarify any part of the problem? Here are five related questions you could consider:

1. How would the probability change if there were more students majoring in other combinations of subjects?
2. What if each couple could sit apart in the seating arrangement problem—how would that affect the probability calculations?
3. How does inclusion-exclusion work for sets involving three or more groups?
4. What if the seating arrangement involved an odd number of people? How would that change the total configurations?
5. Could the seating problem be modeled using graph theory for more complex arrangements?

**Tip:** Always break down complicated set problems with a Venn diagram to visualize how the groups overlap.

1. (4 points) 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, 13 major in Physics, 13 major in Mathematics, 7 major in both Chemistry and Physics, 6 major in both Chemistry and Mathematics, 8 major in both Mathematics and Physics, 5 major in all 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?

2. (4 points) Six people (A, B, C, D, E, and F) forming three couples (A-B, C-D, and E-F) sit on a long bench. The bench has six seats, and all the seating configurations in which the people from the same couple sit next to each other are equally likely.
(a) What is the total number of possible seating configurations (in which the people from each couple sit next to each other) for the six people?
(b) What is the probability that B ends up sitting next to C?

This page solves a complex inclusion-exclusion problem involving students majoring in multiple science subjects, as well as a probability question about seating arrangements of couples. Detailed explanations are provided with step-by-step breakdowns.

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