5. a. Each student in a class of 17 studies at least one subject. Of these, 7 study Art (A), 5 study Biology (B), 12 study Mathematics (M),3 study Art only and 5 study Mathematics only but no one studies both Art and Biology. Find $$ n\left(A^{\prime} \cap M^{\prime} \cap B\right. $$ ) b. Evaluate $$ \sqrt{27}+\sqrt{75}-9 \sqrt{3} $$, correct to three decimal plades. c. Simplify $$ \frac{4}{x+2}+\frac{3}{x+3} $$

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Let's address each part of the problem step by step.

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### **5.a. Find $$ n\left(A^{\prime} \cap M^{\prime} \cap B\right) $$**

**Given:**
- Total students: 17
- Each student studies at least one subject.
- $$ n(A) = 7 $$
- $$ n(B) = 5 $$
- $$ n(M) = 12 $$
- Students studying only Art: 3
- Students studying only Mathematics: 5
- No student studies both Art and Biology.

**Objective:** Find the number of students who do **not** study Art **and** do **not** study Mathematics but **do** study Biology, i.e., $$ n\left(A^{\prime} \cap M^{\prime} \cap B\right) $$.

**Solution:**

1. **Breakdown of Students:**
   - **Art Only (A only):** 3
   - **Mathematics Only (M only):** 5
   - **Both Art and Mathematics (A ∩ M):** Let's denote this as $$ y $$.
   - **Biology Only (B only):** Let's denote this as $$ x $$.
   - **Both Biology and Mathematics (B ∩ M):** Let's denote this as $$ z $$.
   - **No overlap between Art and Biology.**

2. **Using the Given Information:**
   - Total students: $$ 3 $$ (A only) $$ + 5 $$ (M only) $$ + x $$ (B only) $$ + y $$ (A ∩ M) $$ + z $$ (B ∩ M) $$ = 17 $$
   
3. **Breaking Down by Each Subject:**
   - **Art:**
     $$
     n(A) = 7 = 3 \text{ (A only)} + y \Rightarrow y = 4
     $$
   - **Biology:**
     $$
     n(B) = 5 = x \text{ (B only)} + z \text{ (B ∩ M)} \Rightarrow x + z = 5
     $$
   - **Mathematics:**
     $$
     n(M) = 12 = 5 \text{ (M only)} + y \text{ (A ∩ M)} + z \text{ (B ∩ M)} \Rightarrow 12 = 5 + 4 + z \Rightarrow z = 3
     $$
   - Substituting $$ z = 3 $$ into $$ x + z = 5 $$:
     $$
     x + 3 = 5 \Rightarrow x = 2
     $$

**Answer:**
$$
n\left(A^{\prime} \cap M^{\prime} \cap B\right) = 2
$$

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### **5.b. Evaluate $$ \sqrt{27} + \sqrt{75} - 9\sqrt{3} $$ to three decimal places.**

**Solution:**

1. **Simplify the Square Roots:**
   $$
   \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}
   $$
   $$
   \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}
   $$

2. **Combine the Terms:**
   $$
   \sqrt{27} + \sqrt{75} - 9\sqrt{3} = 3\sqrt{3} + 5\sqrt{3} - 9\sqrt{3} = (3 + 5 - 9)\sqrt{3} = -1\sqrt{3} = -\sqrt{3}
   $$

3. **Calculate the Numerical Value:**
   $$
   \sqrt{3} \approx 1.7320508075688772
   $$
   $$
   -\sqrt{3} \approx -1.732
   $$

**Answer:**
$$
\sqrt{27} + \sqrt{75} - 9\sqrt{3} \approx -1.732
$$

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### **5.c. Simplify $$ \frac{4}{x+2} + \frac{3}{x+3} $$**

**Solution:**

1. **Find the Least Common Denominator (LCD):**
   $$
   \text{LCD} = (x + 2)(x + 3)
   $$

2. **Rewrite Each Fraction with the LCD:**
   $$
   \frac{4}{x+2} = \frac{4(x + 3)}{(x + 2)(x + 3)}
   $$
   $$
   \frac{3}{x+3} = \frac{3(x + 2)}{(x + 2)(x + 3)}
   $$

3. **Combine the Fractions:**
   $$
   \frac{4(x + 3)}{(x + 2)(x + 3)} + \frac{3(x + 2)}{(x + 2)(x + 3)} = \frac{4(x + 3) + 3(x + 2)}{(x + 2)(x + 3)}
   $$

4. **Simplify the Numerator:**
   $$
   4(x + 3) + 3(x + 2) = 4x + 12 + 3x + 6 = 7x + 18
   $$

5. **Final Simplified Form:**
   $$
   \frac{7x + 18}{(x + 2)(x + 3)}
   $$

**Answer:**
$$
\frac{4}{x+2} + \frac{3}{x+3} = \frac{7x + 18}{(x + 2)(x + 3)}
$$

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**5.a.** 2 **5.b.** -1.732 **5.c.** $$ \frac{7x + 18}{(x + 2)(x + 3)} $$

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