We can solve this problem using the principle of inclusion and exclusion. Let's define the sets:

• $M$ = set of students who studied Mathematics
• $P$ = set of students who studied Physics
• $C$ = set of students who studied Chemistry

We are given the following information:

• $|M| = 40$ (students who studied Mathematics)
• $|P| = 64$ (students who studied Physics)
• $|C| = 35$ (students who studied Chemistry)
• $|M \cap P \cap C| = 1$ (students who studied all three subjects)
• $|M \cap P| = 25$ (students who studied both Mathematics and Physics)
• $|M \cap C| = 3$ (students who studied both Mathematics and Chemistry)
• $|P \cap C| = 20$ (students who studied both Physics and Chemistry)
• The total number of students is 100.

Step 1: Find the number of students who studied Chemistry only

We can apply the principle of inclusion and exclusion for the students who studied Chemistry.

Using the formula for students who studied Chemistry:

$|C \text{ only}| = |C| - (|M \cap C| + |P \cap C| - |M \cap P \cap C|)$

Substitute the known values:

$|C \text{ only}| = 35 - (3 + 20 - 1) = 35 - 22 = 13$

So, 13 students studied only Chemistry.

Step 2: Find the number of students who studied none of these subjects

The total number of students is 100. To find the number of students who studied none of the three subjects, we need to find the number of students who studied at least one subject and subtract that from 100.

Using the principle of inclusion and exclusion, the formula for students who studied at least one subject is:

$|M \cup P \cup C| = |M| + |P| + |C| - (|M \cap P| + |P \cap C| + |M \cap C|) + |M \cap P \cap C|$

Substitute the known values:

$|M \cup P \cup C| = 40 + 64 + 35 - (25 + 20 + 3) + 1 = 139 - 48 + 1 = 92$

Therefore, 92 students studied at least one subject. The number of students who studied none of these subjects is:

$100 - 92 = 8$

Final Answers:

1. The number of students who studied Chemistry only is 13.
2. The number of students who studied none of these subjects is 8.

Would you like further details on any part of the solution?

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Related Questions:

1. How do we apply the principle of inclusion and exclusion for three sets?
2. What would change if some students studied two subjects but not the third?
3. Can we generalize this method to more than three sets?
4. How would you calculate the number of students who studied exactly two subjects?
5. How can we use Venn diagrams to represent this problem visually?

Tip: The principle of inclusion and exclusion is helpful when dealing with overlapping sets because it avoids double-counting.