In a college team there are 15 players of whoom 3 are teachers . In how many ways can a team of 11 players be selected so as to include only one teachers

Answer

To solve the problem of selecting a team of 11 players from a group of 15 players, including only one teacher, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Players and Teachers:**
- Total players = 15
- Total teachers = 3
- Total non-teachers = 15 - 3 = 12

2. **Select One Teacher:**
- We need to choose 1 teacher from the 3 available teachers. The number of ways to choose 1 teacher from 3 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.
- Thus, the number of ways to choose 1 teacher is:
\[ \binom{3}{1} = 3 \]

3. **Select the Remaining Players:**
- After selecting 1 teacher, we need to choose the remaining players from the non-teachers. Since we are forming a team of 11 players and we have already selected 1 teacher, we need to select 10 players from the 12 non-teachers.
- The number of ways to choose 10 players from 12 is:
\[ \binom{12}{10} = \binom{12}{2} \quad \text{(using the property } \binom{n}{r} = \binom{n}{n-r}\text{)} \]
- Now calculate \( \binom{12}{2} \):
\[ \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \]

4. **Calculate the Total Combinations:**
- Now, multiply the number of ways to choose the teacher by the number of ways to choose the non-teachers:
\[ \text{Total ways} = \binom{3}{1} \times \binom{12}{10} = 3 \times 66 = 198 \]

### Final Answer:
The total number of ways to select a team of 11 players including only one teacher is **198**.