Quintnoid, a hiring agency is looking for young, dynamic professionals to expand its business across the globe. The firm is mainly looking for three qualities viz. Creativity, Patience and Flexibility. The firm called 225 candidates for interview.

The breakup of different candidates with different attributes is shown below in the table.

Attributes

Number of candidates

47

57

96

Patience and Flexible but not Creative

Patience and Creative but not Flexible

Flexible or Creative but not Patience

Each candidate had at least one of the three attributes out of the mentioned three attributes.

The firm could find only three candidates who possessed all the qualities and thus relaxed the criteria a little. It was also found out that for any quality, the number of candidates who had only that quality did not exceed one-fifth of the total number of candidates interviewed

This problem revolves around set theory, particularly using the **principle of inclusion and exclusion (PIE)** to analyze the number of candidates with various combinations of the three qualities: Creativity (C), Patience (P), and Flexibility (F). Let's define some sets based on the given attributes:

- \( C \) = Set of candidates who are **Creative**.
- \( P \) = Set of candidates who are **Patient**.
- \( F \) = Set of candidates who are **Flexible**.

Now, let’s break down the problem into manageable parts based on the provided information:

### Information provided:
1. Total number of candidates: \( |C \cup P \cup F| = 225 \)
2. 47 candidates are **Patient and Flexible but not Creative**: \( |P \cap F \cap \overline{C}| = 47 \)
3. 57 candidates are **Patient and Creative but not Flexible**: \( |P \cap C \cap \overline{F}| = 57 \)
4. 96 candidates are **Flexible or Creative but not Patient**: \( |(F \cup C) \cap \overline{P}| = 96 \)
5. Three candidates possess all three qualities: \( |C \cap P \cap F| = 3 \)
6. For any quality, the number of candidates who had only that quality did not exceed one-fifth of the total number of candidates, i.e., for any quality \( X \in \{C, P, F\} \), \( |X \cap \overline{(Y \cup Z)}| \leq \frac{1}{5} \times 225 = 45 \).

### Goals:
- We need to find the number of candidates in each relevant category based on the intersection of the sets \( C \), \( P \), and \( F \).

### Step-by-Step Solution Using Inclusion-Exclusion:

Let’s define the relevant groups:
- Candidates possessing only one attribute:
  - \( |C \cap \overline{(P \cup F)}| \): Only Creative.
  - \( |P \cap \overline{(C \cup F)}| \): Only Patient.
  - \( |F \cap \overline{(C \cup P)}| \): Only Flexible.
  
- Candidates possessing exactly two attributes:
  - \( |C \cap P \cap \overline{F}| = 57 \)
  - \( |P \cap F \cap \overline{C}| = 47 \)
  - \( |F \cap C \cap \overline{P}| \)

- Candidates possessing all three attributes:
  - \( |C \cap P \cap F| = 3 \)

- Total: \( |C \cup P \cup F| = 225 \)

### Using the given conditions:
1. The number of candidates possessing only one attribute cannot exceed 45, meaning:
   - \( |C \cap \overline{(P \cup F)}| \leq 45 \)
   - \( |P \cap \overline{(C \cup F)}| \leq 45 \)
   - \( |F \cap \overline{(C \cup P)}| \leq 45 \)

### Next steps:
We can use inclusion-exclusion to calculate the total number of candidates:
\[
|C \cup P \cup F| = |C| + |P| + |F| - |C \cap P| - |P \cap F| - |F \cap C| + |C \cap P \cap F|
\]
Now we need to solve for the unknown quantities by plugging the values given and using the information about candidates possessing combinations of attributes.

Would you like me to continue with the detailed calculations, or do you have any specific questions at this stage? Also, here are 5 related questions for further exploration:

1. What is the number of candidates who possess only Creativity?
2. How many candidates are only Patient?
3. What is the breakdown of candidates who possess exactly two attributes?
4. How does the principle of inclusion-exclusion work for solving such problems?
5. Can we generalize this method to larger sets with more attributes?

**Tip:** When working with problems involving sets, always draw a Venn diagram to visualize intersections and unions. It helps in organizing the information and applying the inclusion-exclusion principle.

Quintnoid, a hiring agency is looking for young, dynamic professionals to expand its business across the globe. The firm is mainly looking for three qualities viz. Creativity, Patience and Flexibility. The firm called 225 candidates for interview. The breakup of different candidates with different attributes is shown below in the table. Attributes Number of candidates: 47 (Patience and Flexible but not Creative), 57 (Patience and Creative but not Flexible), 96 (Flexible or Creative but not Patience). Each candidate had at least one of the three attributes out of the mentioned three attributes. The firm could find only three candidates who possessed all the qualities and thus relaxed the criteria a little. It was also found out that for any quality, the number of candidates who had only that quality did not exceed one-fifth of the total number of candidates interviewed.

This problem involves analyzing the number of candidates possessing various attributes (Creativity, Patience, and Flexibility) using the principle of inclusion-exclusion. Learn how to break down the problem and apply set theory to solve complex candidate selection challenges, suitable for advanced high school or college-level students.

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