https://file.aiquickdraw.com/mathbot/file/17334316175510rgx5jo7.jpg

The question reads:

> In a class of 8 boys and 6 girls, 7 students are to be selected for a tuition fee scholarship. In how many ways can this be done if there must be exactly 3 boys?

### Solution:

We need to select exactly **3 boys** from 8 boys and **4 girls** (since \( 7 - 3 = 4 \)) from 6 girls.

1. **Number of ways to select 3 boys out of 8:**

\[
\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56
\]

2. **Number of ways to select 4 girls out of 6:**

\[
\binom{6}{4} = \binom{6}{2} = \frac{6!}{4!(6-4)!} = \frac{6 \cdot 5}{2 \cdot 1} = 15
\]

3. **Total number of combinations:**

The total number of ways to select 3 boys and 4 girls is:

\[
\binom{8}{3} \cdot \binom{6}{4} = 56 \cdot 15 = 840
\]

### Final Answer:
There are **840 ways** to select 7 students with exactly 3 boys.

---

Would you like further clarification or a breakdown of the steps? Here are related questions for further understanding:

1. How is the combination formula derived and why is it used here?  
2. How do we compute \(\binom{n}{r}\) for larger values of \(n\) and \(r\)?  
3. What if the condition was at least 3 boys instead of exactly 3?  
4. How does the problem change if the class size is different?  
5. Can we calculate this using a probability-based approach instead of combinations?

**Tip:** Always verify whether "order matters" in selection problems. If it does, use permutations instead of combinations!

In a class of 8 boys and 6 girls, 7 students are to be selected for a tuition fee scholarship. In how many ways can this be done if there must be exactly 3 boys?

Solve a combinatorics problem involving the selection of 7 students from a group of 8 boys and 6 girls for a scholarship, ensuring exactly 3 boys. Explore the combination formula and key principles of combinatorics in this detailed step-by-step solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation