Understanding the Group Survey Problem

The problem provides us with data about the preferences of a group of 75 students regarding four different vegetables: cabbage, cauliflower, carrot, and capsicum. We are given specific numbers of students who like certain combinations of these vegetables, including those who like only one particular vegetable.

The question asks us to find the total number of students who like only one vegetable.

Breaking Down the Vegetable Preference Data

Let's list the information provided about the students' preferences:

• Total number of students: 75
• Number of students who like only cabbage: 12
• Number of students who like only cauliflower: 15
• Number of students who like only carrot: 21
• Number of students who like both carrot and cabbage: 12
• Number of students who like only capsicum: 13
• Number of students who like both capsicum and cauliflower: 2

Identifying Students Liking Only One Vegetable

The question specifically asks for students who like only one vegetable. This means we need to look at the numbers provided for students who like just one type of vegetable and no other. From the given data, the following groups represent students who like only one vegetable:

• Students who like only cabbage.
• Students who like only cauliflower.
• Students who like only carrot.
• Students who like only capsicum.

The information about students liking "both carrot and cabbage" or "both capsicum and cauliflower" refers to students who like more than one vegetable, so these numbers are not included when calculating the total for those liking only one vegetable.

Calculating the Total Number of Students Liking Only One Vegetable

To find the total number of students who like only one vegetable, we simply need to sum the counts for each group that likes only one specific vegetable.

Number of students liking only one vegetable = (Students liking only cabbage) + (Students liking only cauliflower) + (Students liking only carrot) + (Students liking only capsicum)

Substituting the given values:

\( \text{Number of students liking only one vegetable} = 12 + 15 + 21 + 13 \)

Performing the Summation

Let's add the numbers:

• \( 12 \text{ (only cabbage)} \)
• \( 15 \text{ (only cauliflower)} \)
• \( 21 \text{ (only carrot)} \)
• \( 13 \text{ (only capsicum)} \)

Sum \( = 12 + 15 + 21 + 13 = 27 + 21 + 13 = 48 + 13 = 61 \)

So, a total of 61 students like only one vegetable.

Conclusion

Based on the calculation, 61 students like only one vegetable. We identified the relevant data points from the problem statement and summed them up to arrive at the answer.

Revision Table: Vegetable Preferences

Additional Information: Survey Data Analysis

Problems like this often involve analyzing survey data or set theory concepts. Understanding the phrasing is key:

• "Only X": Refers exclusively to liking item X and no other items mentioned.
• "Both X and Y": Typically refers to liking item X and item Y, possibly also others, unless specified as "only X and Y". In this problem, the explicit use of "only" for single vegetables suggests "both" without "only" means liking at least those two. However, for the specific question asking for "only one vegetable", only the "only X" categories are relevant.
• Venn Diagrams can be helpful for visualizing overlaps and distinct groups in more complex survey problems involving multiple sets.

This question was simpler as it directly provided the numbers for the groups that like only one vegetable, making the calculation a direct summation.