https://file.aiquickdraw.com/mathbot/file/17296265366520wcoubta.jpg

From the image provided, the task is based on a **weighted voting system** with the structure \([47: 10, 9, 9, 5, 4, 4, 3, 2, 2]\). Let's go through the questions:

### 1. How many players are there?
The players correspond to the weights provided in the voting system.

- The weights given are: \(10, 9, 9, 5, 4, 4, 3, 2, 2\).

Thus, there are **9 players** in total.

### 2. What is the total number (weight) of votes?
The total number of votes is the sum of the weights of all the players:

\[
10 + 9 + 9 + 5 + 4 + 4 + 3 + 2 + 2 = 48
\]

So, the total number (or weight) of votes is **48**.

### 3. What is the quota in this system?
The quota is the minimum number of votes needed to win the vote. In this case, the quota is given as the first number in the system notation, which is **47**.

### Summary:
- Number of players: **9**
- Total weight of votes: **48**
- Quota: **47**

Let me know if you'd like more details on any part! 

Here are 5 related questions to further explore the topic:
1. What is the significance of having a quota in a weighted voting system?
2. How would the results change if the quota was lower than 47?
3. Can a player with a smaller weight (e.g., 2) still be pivotal in any coalition?
4. How would you calculate the Banzhaf power index for each player?
5. What does it mean if no coalition can meet the quota?

**Tip**: Always ensure the sum of weights exceeds the quota by a small margin to avoid deadlock in a weighted voting system.

Consider the weighted voting system [47: 10, 9, 9, 5, 4, 4, 3, 2, 2].
a. How many players are there?
b. What is the total number (weight) of votes?
c. What is the quota in this system?

Explore the solution to a weighted voting system problem with a quota of 47 and 9 players. Understand how to calculate the total weight of votes and the significance of quotas in such systems. Suitable for Grades 10-12.

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