A team without both
 Let's say two members of a group don't get along and refuse to work together as a team.  Find the number of formats in which teams can be organized according to these conditions?

To solve the problem of determining how many ways teams can be formed given that two specific members refuse to work together, we can approach it by calculating the total number of possible teams and then subtracting the number of teams where both members are included together.

### Let's define the situation:
- Let the total number of members in the group be \( n \).
- Let the two specific members who refuse to work together be \( A \) and \( B \).

### Step 1: Calculate the total number of possible teams.
For any member in a group, there are two options: either they are included in a team or they are not. Therefore, the total number of possible team formations (including the empty set) is \( 2^n \).

### Step 2: Subtract cases where both members \( A \) and \( B \) are on the same team.
- If both \( A \) and \( B \) are included in a team, we are left with the other \( n-2 \) members. 
- The number of teams that can be formed with the remaining \( n-2 \) members is \( 2^{n-2} \).

### Step 3: Find the number of valid teams.
The number of valid teams (where \( A \) and \( B \) are not together) is given by:
\[
2^n - 2^{n-2}
\]
This expression accounts for all possible teams and removes the ones that include both members \( A \) and \( B \).

### Final Answer:
Thus, the number of possible team formations under the given condition is:
\[
\boxed{2^n - 2^{n-2}}
\]

This is the number of ways to form teams where members \( A \) and \( B \) do not work together.

Do you want more details or have any questions?

Here are some related questions to explore:
1. How does the team formation change if three members refuse to work together?
2. What if members \( A \) and \( B \) must be on the same team instead of not together?
3. How does this formula change if the total number of members \( n \) is known?
4. What happens if we need to form teams of a specific size, excluding \( A \) and \( B \)?
5. Can this problem be solved using combinatorics?

**Tip:** Understanding how to include or exclude specific cases in counting problems is a powerful technique in combinatorics known as the "Inclusion-Exclusion Principle."

Learn how to count the number of team formations where two specific members refuse to work together using combinatorial principles and the inclusion-exclusion principle. This problem involves calculating the total number of teams and subtracting cases where both specified members are together.

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