A survey of 40 students showed that 23 students study Mathematics, 5 study Mathematics and Physics, 8 study Chemistry and Mathematics, 5 study Physics and C...

Set notation and the Venn diagram

The numbers studying two subjects include the 3 students who study all three subjects. Therefore, subtract 3 from each two-subject total to find the regions for exactly two subjects.

Finding the regions

• Mathematics and Physics only: \(5-3=2\).
• Mathematics and Chemistry only: \(8-3=5\).
• Physics and Chemistry only: \(5-3=2\).

There are 23 students studying Mathematics in total, so:

\[ \text{Mathematics only}=23-(2+5+3)=13. \]

Let the number studying Chemistry only be \(x\). The question states that Physics only is twice this number, so Physics only is \(2x\).

Using the intended interpretation that all 40 surveyed students are in the three-subject Venn diagram:

\[13+2+5+2+3+x+2x=40\]

\[25+3x=40\]

\[3x=15,\qquad x=5.\]

Therefore:

• Only Physics \(=2x=2(5)=\mathbf{10}\) students.
• Only one subject \(=13+10+5=\mathbf{28}\) students.

Students who study exactly two subjects are in the three pair-only regions:

\[2+5+2=9.\]

Hence the required probability is:

\[P(\text{exactly two subjects})=\frac{9}{40}=0.225.\]

The denominator must be \(40\), the total number of surveyed students. Although the supplied reference includes \(\frac{9}{48}\), this is inconsistent with both the survey total and the decimal \(0.225\); \(\frac{9}{40}\) is the correct probability.

Examination reminder: When a total for two subjects includes students studying all three, subtract the all-three region before counting students who study exactly two subjects.