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The problem asks us to find the number of ways to choose one teacher, one girl, and one boy from a school with a total of 16 teachers and 220 students. The ratio of girls to boys is 6:5.

Let's break this down step by step:

### Step 1: Determine the number of girls and boys
- The total number of students is 220.
- The ratio of girls to boys is \(6:5\). This means for every 6 girls, there are 5 boys.
  
Let the number of girls be \(6x\) and the number of boys be \(5x\), where \(x\) is a common factor.

The total number of students is given by:
\[
6x + 5x = 220
\]
\[
11x = 220
\]
\[
x = \frac{220}{11} = 20
\]

Now, we can calculate the number of girls and boys:
- Number of girls = \(6x = 6 \times 20 = 120\)
- Number of boys = \(5x = 5 \times 20 = 100\)

### Step 2: Calculate the number of ways to choose one teacher, one girl, and one boy
- There are 16 teachers, 120 girls, and 100 boys.

To find the total number of ways to choose one teacher, one girl, and one boy, we multiply the number of choices for each:
\[
\text{Total number of ways} = 16 \times 120 \times 100
\]

\[
\text{Total number of ways} = 16 \times 12,000 = 192,000
\]

### Conclusion
The number of different ways to choose one teacher, one girl, and one boy is **192,000**.

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Would you like more details on any of the steps? Here are some related questions for further understanding:

1. How does the ratio \(6:5\) help in solving the problem?
2. Can we use a different method to calculate the number of girls and boys?
3. What would happen if the ratio of girls to boys changed?
4. How can this concept of combinations be applied to other scenarios?
5. Can you explain how combinations differ from permutations?

**Tip:** When dealing with ratios, always convert them into equations involving a common variable to solve for unknown quantities.

In a school, there are 16 teachers and 220 students. The ratio of girls to boys in the school is 6:5. One teacher, one girl, and one boy are going to be chosen to represent the school. Work out the number of different ways there are to choose one teacher, one girl, and one boy.

Solve a combinatorics problem involving 16 teachers, 220 students with a 6:5 ratio of girls to boys. Learn how to determine the number of ways to select one teacher, one girl, and one boy to represent the school. Suitable for Grades 7-9.

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