Suppose that a weighted voting system has 9 players. How many coalitions are possible? 


To determine how many coalitions are possible in a weighted voting system with 9 players, we need to recognize that a coalition is simply a subset of the total set of players. Each player can either be included or not included in a coalition, meaning each player has 2 choices: to be part of the coalition or not.

For \( n \) players, the number of coalitions (including the empty coalition where no players are involved) is given by:

\[
2^n
\]

In this case, \( n = 9 \), so the number of possible coalitions is:

\[
2^9 = 512
\]

Thus, the number of possible coalitions is **512**.

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How many non-empty coalitions are there with 9 players?
2. What is a winning coalition in a weighted voting system?
3. If each player has a different weight, how does this affect the coalitions?
4. What is the Banzhaf power index in weighted voting?
5. How many coalitions are there if 3 out of 9 players must always be in the coalition?

**Tip:** In voting systems, the power of each player can differ significantly based on the voting weights and not just the number of coalitions they are part of.

Suppose that a weighted voting system has 9 players. How many coalitions are possible?

Explore how to calculate the number of possible coalitions in a weighted voting system with 9 players. Using combinatorics and set theory, we find that there are 512 possible coalitions, including the empty coalition. This problem is suitable for students in Grades 9-12.

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