A survey of 145
 freshmen business students at a local university produced the results listed below. How many students took English and math, but not music?

28
 took English;
36
 took math;
31
 took music;
19
 took English but not math;
14
 took math and music;
12
 took English and music;
6
 took all three

Let's solve this using set theory and a Venn diagram. 

We are given the following information:

- Total freshmen surveyed: \( 145 \)
- Students who took English: \( E = 28 \)
- Students who took math: \( M = 36 \)
- Students who took music: \( Mu = 31 \)
- Students who took English but not math: \( E \cap \overline{M} = 19 \)
- Students who took math and music: \( M \cap Mu = 14 \)
- Students who took English and music: \( E \cap Mu = 12 \)
- Students who took all three subjects: \( E \cap M \cap Mu = 6 \)

We are asked to find how many students took **English and math, but not music**. This is represented as \( E \cap M \cap \overline{Mu} \).

### Step-by-step breakdown:

1. **Number of students who took all three subjects** is already given as \( E \cap M \cap Mu = 6 \).

2. **Number of students who took math and music** is \( M \cap Mu = 14 \), but this includes those who took all three. So, the number of students who took only math and music is:
   \[
   (M \cap Mu) - (E \cap M \cap Mu) = 14 - 6 = 8
   \]

3. **Number of students who took English and music** is \( E \cap Mu = 12 \), but this includes those who took all three. So, the number of students who took only English and music is:
   \[
   (E \cap Mu) - (E \cap M \cap Mu) = 12 - 6 = 6
   \]

4. **Number of students who took English but not math** is given as \( E \cap \overline{M} = 19 \).

5. Now, let's calculate the total number of students who took both English and math. This is represented as \( E \cap M \), but it includes students who took all three, as well as those who took only English and math. We don't know \( E \cap M \cap \overline{Mu} \) directly, so let's find it:

   The total number of students who took English and math is:
   \[
   (E \cap M) = E - (E \cap \overline{M}) = 28 - 19 = 9
   \]
   Out of these 9 students, 6 took all three subjects, so the number of students who took only English and math (but not music) is:
   \[
   E \cap M \cap \overline{Mu} = (E \cap M) - (E \cap M \cap Mu) = 9 - 6 = 3
   \]

Thus, **3 students took English and math, but not music**.

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Would you like more details or have any questions about this problem?

### Related questions:
1. How many students took only math?
2. How many students took only music?
3. How many students took only English?
4. How many students took exactly two subjects?
5. How many students took none of the three subjects?

### Tip:
When dealing with overlapping sets, Venn diagrams or set notations are helpful for organizing information and solving step by step.

A survey of 145 freshmen business students at a local university produced the results listed below. How many students took English and math, but not music? 28 took English; 36 took math; 31 took music; 19 took English but not math; 14 took math and music; 12 took English and music; 6 took all three.

Solve a set theory problem involving 145 freshmen. Determine how many students took English and math but not music using Venn diagrams and the inclusion-exclusion principle. This problem explores concepts of set theory and is suitable for students in Grades 9-12.

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