Understanding the Student Class Data

This problem involves analyzing the number of students taking different music classes: guitar, drum, and violin. We are given information about the total number of students, the number of students taking at least one class, students taking specific combinations of classes, and students taking only one specific class. Our goal is to find the number of students who take part only in guitar classes.

We can approach this problem using the principles of set theory and Venn diagrams, even without drawing the diagram explicitly. We'll break down the student population into disjoint groups based on the classes they take.

Given Information Breakdown

Let's list the information provided in the question:

• Total number of students in the class = 170
• Number of students taking at least one class (Guitar, Drum, or Violin) = 155
• Number of students taking part in any two of the classes (exactly two) = 21
• Number of students taking part only in drum classes = 54
• Number of students taking part only in violin classes = 43
• Number of students taking part in all three classes = 5

We need to find the number of students who take part only in guitar classes.

Applying Set Theory Logic

The total number of students taking at least one class is the sum of students in all the disjoint regions of the Venn diagram:

• Students taking only Guitar
• Students taking only Drum
• Students taking only Violin
• Students taking Guitar and Drum only
• Students taking Guitar and Violin only
• Students taking Drum and Violin only
• Students taking all three classes (Guitar, Drum, and Violin)

Let:

• \( N(G_{only}) \) = Number of students taking only Guitar classes
• \( N(D_{only}) \) = Number of students taking only Drum classes = 54
• \( N(V_{only}) \) = Number of students taking only Violin classes = 43
• \( N(G \cap D_{only}) \) = Number of students taking Guitar and Drum classes only
• \( N(G \cap V_{only}) \) = Number of students taking Guitar and Violin classes only
• \( N(D \cap V_{only}) \) = Number of students taking Drum and Violin classes only
• \( N(G \cap D \cap V) \) = Number of students taking all three classes = 5

The number of students taking "any two of the classes" refers to those taking exactly two classes. So,

\[ N(G \cap D_{only}) + N(G \cap V_{only}) + N(D \cap V_{only}) = 21 \]

The total number of students taking at least one class is the sum of all these parts:

\[ \text{Total students (at least one class)} = N(G_{only}) + N(D_{only}) + N(V_{only}) + N(G \cap D_{only}) + N(G \cap V_{only}) + N(D \cap V_{only}) + N(G \cap D \cap V) \]

Calculating Students Only in Guitar Classes

Substitute the known values into the equation:

\[ 155 = N(G_{only}) + 54 + 43 + (N(G \cap D_{only}) + N(G \cap V_{only}) + N(D \cap V_{only})) + 5 \]

We know that \( N(G \cap D_{only}) + N(G \cap V_{only}) + N(D \cap V_{only}) = 21 \). Substitute this into the equation:

\[ 155 = N(G_{only}) + 54 + 43 + 21 + 5 \]

Now, sum the known numbers on the right side:

\[ 54 + 43 = 97 \]

\[ 97 + 21 = 118 \]

\[ 118 + 5 = 123 \]

So the equation becomes:

\[ 155 = N(G_{only}) + 123 \]

To find \( N(G_{only}) \), subtract 123 from 155:

\[ N(G_{only}) = 155 - 123 \]

\[ N(G_{only}) = 32 \]

Therefore, 32 students take part only in guitar classes.

Conclusion

By breaking down the total number of students taking at least one music class into mutually exclusive groups and using the given information, we calculated that the number of students taking part only in guitar classes is 32.

Revision Table: Music Class Students

Additional Information: Venn Diagram Concepts

Venn diagrams are visual tools used to represent the relationships between different sets. In problems like this involving student preferences for classes, they help organize the data.

• Universal Set: The entire group of students (170 in this case).
• Sets: The groups of students taking specific classes (Guitar, Drum, Violin).
• Union (\(\cup\)): Students in at least one set (G \(\cup\) D \(\cup\) V). This is the total number of students involved in the activities.
• Intersection (\(\cap\)): Students common to multiple sets (e.g., G \(\cap\) D for students taking both Guitar and Drum).
• Disjoint Regions: The non-overlapping parts of the Venn diagram representing students taking *only* one specific class, *exactly* two specific classes, or *all three* classes. The sum of students in all these disjoint regions within the sets equals the number of students in the union.

The problem gives us the size of the union (155) and the sizes of several disjoint regions or combinations of disjoint regions (Only Drum, Only Violin, Exactly Two, All Three). By subtracting the known parts from the total size of the union, we can find the size of the remaining disjoint region, which is 'Only Guitar'.