There are 5 candidates for the EVP position and 6 candidates for the VP-Academics position in a certain student organization. Five candidates are listed in the VP-Activities, but closer inspection shows that there are 2 individuals listed as candidates in all three positions. Three individuals are running for both VP Activities but not for EVP. Two candidates are vying for the EVP position alone. What is the total number of individuals running for the officerships?

To solve this problem, let's analyze the information step by step and calculate the total number of distinct individuals running for officerships.

### Given Information:
1. There are 5 candidates for the EVP position.
2. There are 6 candidates for the VP-Academics position.
3. There are 5 candidates for the VP-Activities position.
4. 2 individuals are listed as candidates in all three positions.
5. 3 individuals are running for both VP-Activities and VP-Academics but not for EVP.
6. 2 candidates are running for the EVP position alone.

### Breakdown:

1. **Total number of candidates in EVP**: 
   - There are 5 candidates in total for EVP.
   - Out of these, 2 are running for all three positions.
   - 2 are running for EVP alone.
   
   Therefore, the 5th candidate is running for EVP and either VP-Academics or VP-Activities, but it isn't explicitly mentioned which one.

2. **Total number of candidates in VP-Academics**: 
   - There are 6 candidates in total for VP-Academics.
   - 2 of these candidates are running for all three positions.
   - 3 are running for both VP-Academics and VP-Activities but not for EVP.

   Therefore, the 6th candidate is running for VP-Academics alone.

3. **Total number of candidates in VP-Activities**:
   - There are 5 candidates in total for VP-Activities.
   - 2 of these candidates are running for all three positions.
   - 3 are running for both VP-Activities and VP-Academics but not for EVP.

### Counting Unique Individuals:

- **Candidates running for all three positions**: 2 individuals.
- **Candidates running only for EVP**: 2 individuals.
- **Candidates running for VP-Academics alone**: 1 individual (from the 6 total VP-Academics candidates minus the 2 in all three positions and 3 in both VP-Academics and VP-Activities).
- **Candidates running for VP-Academics and VP-Activities but not EVP**: 3 individuals.
- **The last candidate running for EVP and VP-Academics or VP-Activities**: 1 individual.

Adding these up:

\[
2 \text{ (all three positions)} + 2 \text{ (EVP alone)} + 1 \text{ (VP-Academics alone)} + 3 \text{ (VP-Academics and VP-Activities but not EVP)} + 1 \text{ (either EVP and VP-Academics or VP-Activities)} = 9
\]

So, there are **9 distinct individuals** running for the officerships.

Would you like more details or have any questions?

### Further Questions:
1. What strategies can be used to verify the distinct count of candidates in such a problem?
2. How does the inclusion-exclusion principle apply in counting problems like this?
3. Can you provide a diagram to visualize the overlap between candidates?
4. What are other typical problems where similar counting methods are used?
5. How would the total change if one more candidate joined each position without overlap?

**Tip:** For counting problems, using Venn diagrams can help visualize the overlaps and make the solution more intuitive.

Explore how to count candidates in officer positions using set theory and the inclusion-exclusion principle. Solve a problem involving candidates running for multiple student organization positions with detailed steps and analysis.

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