Answer these questions based on the following information.
There are twelve security guards made to stand around a circular football stadium. Any six guards who are adjacent to each other, need a replacement. In how many ways can these guards be selected for replacement?

Options

A. 36
B. 18
C. 2
D. 12

Answer

To solve the problem of selecting 6 adjacent security guards from a total of 12 standing around a circular football stadium, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arrangement**:
- We have 12 security guards arranged in a circle. The guards can be numbered from 1 to 12.

2. **Identifying Groups of 6**:
- Since we need to select 6 guards who are adjacent to each other, we can visualize this as selecting a block of 6 guards.

3. **Counting the Possible Groups**:
- We can start counting the groups of 6 guards. The first group can be guards 1 to 6. The next group can be guards 2 to 7, and so on.
- Continuing this way, we can form the following groups:
- Group 1: Guards 1, 2, 3, 4, 5, 6
- Group 2: Guards 2, 3, 4, 5, 6, 7
- Group 3: Guards 3, 4, 5, 6, 7, 8
- Group 4: Guards 4, 5, 6, 7, 8, 9
- Group 5: Guards 5, 6, 7, 8, 9, 10
- Group 6: Guards 6, 7, 8, 9, 10, 11
- Group 7: Guards 7, 8, 9, 10, 11, 12
- Group 8: Guards 8, 9, 10, 11, 12, 1
- Group 9: Guards 9, 10, 11, 12, 1, 2
- Group 10: Guards 10, 11, 12, 1, 2, 3
- Group 11: Guards 11, 12, 1, 2, 3, 4
- Group 12: Guards 12, 1, 2, 3, 4, 5

4. **Total Count of Groups**:
- We can see that there are a total of 12 different groups of 6 adjacent guards that can be formed.

5. **Conclusion**:
- Therefore, the total number of ways to select 6 adjacent guards for replacement is **12**.

### Final Answer:
The number of ways to select 6 adjacent guards for replacement is **12**.

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