I. Draw a Venn diagram to represent this information. Use your Venn diagram to calculate the number of students who take all three subjects.

The number of students taking at least one subject is the total number of students minus those taking none.

$$|M \cup A \cup C| = N - (\text{students taking none})$$ $$|M \cup A \cup C| = 40 - 9$$ $$|M \cup A \cup C| = 31$$

The formula is:

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

Substitute the known values into the formula:

$$31 = 19 + 13 + 17 - (8 + 7 + 9) + |M \cap A \cap C|$$ $$31 = 49 - 24 + |M \cap A \cap C|$$ $$31 = 25 + |M \cap A \cap C|$$

$$|M \cap A \cap C| = 31 - 25$$ $$|M \cap A \cap C| = 6$$

So, 6 students take all three subjects.

Let $x = |M \cap A \cap C| = 6$.

• Students taking only Mathematics and Agriculture: $|M \cap A| - x = 8 - 6 = 2$
• Students taking only Agriculture and Chichewa: $|A \cap C| - x = 7 - 6 = 1$
• Students taking only Chichewa and Mathematics: $|C \cap M| - x = 9 - 6 = 3$
• Students taking only Mathematics: $|M| - (\text{only M and A}) - (\text{only C and M}) - x = 19 - 2 - 3 - 6 = 8$
• Students taking only Agriculture: $|A| - (\text{only M and A}) - (\text{only A and C}) - x = 13 - 2 - 1 - 6 = 4$
• Students taking only Chichewa: $|C| - (\text{only A and C}) - (\text{only C and M}) - x = 17 - 1 - 3 - 6 = 7$

A Venn diagram would consist of a rectangle representing the universal set of 40 students. Inside this rectangle, there would be three overlapping circles labeled M (Mathematics), A (Agriculture), and C (Chichewa). The numbers in each region would be:

• The intersection of all three circles ($M \cap A \cap C$): 6 students.
• The region for $M \cap A$ only (not including C): 2 students.
• The region for $A \cap C$ only (not including M): 1 student.
• The region for $C \cap M$ only (not including A): 3 students.
• The region for M only (not A or C): 8 students.
• The region for A only (not M or C): 4 students.
• The region for C only (not M or A): 7 students.
• The region outside all three circles (students taking none): 9 students.

To verify, sum all regions: $6 + 2 + 1 + 3 + 8 + 4 + 7 + 9 = 40$. This matches the total number of students.

The number of students who take all three subjects is $\text{6 students}$.