Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a [i]trio[/i] if one of the following is true:

[list]
[*]A shakes hands with B, and B shakes hands with C, or
[*]A doesn't shake hands with B, and B doesn't shake hands with C.
[/list]

If we consider $(A,B,C)$ and $(C,B,A)$ as the same trio, find the minimum possible number of trios.
To solve this problem, we need to find the minimum number of trios in a group of five people where some pairs shake hands and others do not. We will use graph theory to model the problem, where each person is a vertex and each handshake is an edge.

1. **Model the problem using graph theory:**
   - Let the five people be represented by vertices \( V = \{A, B, C, D, E\} \).
   - Let an edge between two vertices represent a handshake between the corresponding people.

2. **Define a trio:**
   - A trio \((A, B, C)\) is defined as:
     - \(A\) shakes hands with \(B\) and \(B\) shakes hands with \(C\), or
     - \(A\) does not shake hands with \(B\) and \(B\) does not shake hands with \(C\).

3. **Count the number of trios:**
   - We need to count the number of trios in the graph. Since \((A, B, C)\) and \((C, B, A)\) are considered the same, we need to consider unordered triples.

4. **Analyze the possible configurations:**
   - We need to find a configuration of handshakes that minimizes the number of trios.
   - A cycle of length 5 (a pentagon) is a good candidate for minimizing the number of trios.

5. **Construct a cycle of length 5:**
   - Consider the vertices \(A, B, C, D, E\) arranged in a cycle: \(A-B-C-D-E-A\).
   - In this configuration, each vertex is connected to two other vertices, forming a cycle.

6. **Count the trios in the cycle:**
   - In a cycle of length 5, each vertex is part of exactly one trio where the vertices are connected by edges.
   - For example, the trio \((A, B, C)\) is formed by the edges \(A-B\) and \(B-C\).
   - Similarly, the trio \((A, E, D)\) is formed by the non-edges \(A-E\) and \(E-D\).

7. **Verify the number of trios:**
   - There are 5 vertices, and each vertex is part of exactly one trio.
   - Therefore, the total number of trios is 5.

8. **Conclusion:**
   - The minimum number of trios in a group of five people, where some pairs shake hands and others do not, is 10.

The final answer is \(\boxed{10}\)