A group of 70 students were surveyed about the subjects they offer: English (E), Mathematics (M), and Geography (G). The following information was collected: 26 students offer all three subjects, 9 of

Step 7: Calculate $n(M \text{ only})$.
All students offer at least one subject, so the sum of all regions must equal the total number of students.

$$n(E \cup M \cup G) = a + b + c + d + e + f + g = 70$$

Substitute the values found: $a=6, c=4, d=9, e=12, f=8, g=26$.

$$6 + b + 4 + 9 + 12 + 8 + 26 = 70$$ $$b + 65 = 70$$ $$b = 70 - 65$$ $$b = 5$$

So, $n(M \text{ only}) = 5$.

The number of students in each region of the Venn diagram is:

• $n(E \text{ only}) = 6$
• $n(M \text{ only}) = 5$
• $n(G \text{ only}) = 4$
• $n(E \cap M \text{ only}) = 9$
• $n(E \cap G \text{ only}) = 12$
• $n(M \cap G \text{ only}) = 8$
• $n(E \cap M \cap G) = 26$

a) Illustrate the information in a Venn diagram.
A Venn diagram with three intersecting circles (E, M, G) would show the following values in their respective regions:

• Region E only: 6
• Region M only: 5
• Region G only: 4
• Region E and M only (not G): 9
• Region E and G only (not M): 12
• Region M and G only (not E): 8
• Region E, M, and G (all three): 26

b) Determine the number of students who offer:

i) mathematics
The number of students who offer Mathematics is the sum of all regions within the Mathematics circle.

$$n(M) = n(M \text{ only}) + n(E \cap M \text{ only}) + n(M \cap G \text{ only}) + n(E \cap M \cap G)$$ $$n(M) = b + d + f + g$$ $$n(M) = 5 + 9 + 8 + 26$$ $$n(M) = 48$$

The number of students who offer mathematics is $\boxed{\text{48}}$.

ii) mathematics and economics but not geography
This corresponds to the region where Economics and Mathematics circles overlap, but outside the Geography circle. This is represented by $d$.

$$n(M \cap E \cap G^c) = d$$ $$n(M \cap E \cap G^c) = 9$$

The number of students who offer mathematics and economics but not geography is $\boxed{\text{9}}$.

iii) The probability that a student selected at random from the class offer geography only.
The number of students who offer Geography only is $c = 4$.
The total number of students in the class is $n(U) = 70$.
The probability is given by:

$$P(\text{Geography only}) = \frac{\text{Number of students who offer Geography only}}{\text{Total number of students}}$$ $$P(\text{Geography only}) = \frac{c}{n(U)}$$ $$P(\text{Geography only}) = \frac{4}{70}$$

Simplify the fraction:

$$P(\text{Geography only}) = \frac{2}{35}$$

The probability that a student selected at random from the class offers geography only is $\boxed{\frac{2}{35}}$.