To find the number of candidates who are proficient in exactly one of the three programming languages, we will use the principle of inclusion-exclusion and some set theory.

Let's denote the following:

• $P$ as the set of candidates proficient in Python.
• $C$ as the set of candidates proficient in C.
• $J$ as the set of candidates proficient in Java.

Given dataTo find the number of candidates who are proficient in exactly one of the three programming languages, we will use the principle of inclusion-exclusion and some set theory.

Let's denote the following:

• $P$ as the set of candidates proficient in Python.
• $C$ as the set of candidates proficient in C.
• $J$ as the set of candidates proficient in Java.

Given

• Total candidates ($n$) = 250
• Candidates proficient in Python ($|P|$) = 105
• Candidates proficient in C ($|C|$) = 95
• Candidates proficient in Java ($|J|$) = 115
• Candidates proficient in Python and Java ($|P \cap J|$) = 15
• Candidates proficient in C and Python ($|C \cap P|$) = 50
• Candidates proficient in C and Java ($|C \cap J|$) = 25
• Candidates who don't know any of the programming languages = 10

First, let's calculate the number of candidates who know at least one programming language: $|P \cup C \cup J| = n - \text{Candidates who don't know any programming languages} = 250 - 10 = 240$

Using the principle of inclusion-exclusion to find $|P \cup C \cup J|$: $|P \cup C \cup J| = |P| + |C| + |J| - |P \cap C| - |C \cap J| - |P \cap J| + |P \cap C \cap J|$

We need $|P \cap C \cap J|$. Rearrange the inclusion-exclusion principle to solve for $|P \cap C \cap J|$: $240 = 105 + 95 + 115 - 50 - 25 - 15 + |P \cap C \cap J|$ $240 = 225 + |P \cap C \cap J|$ $|P \cap C \cap J| = 240 - 225 = 15$

Now, we calculate the number of candidates proficient in exactly one language. This is done by subtracting the number of candidates proficient in multiple languages from each total:

• Candidates proficient only in Python ($|P \setminus (C \cup J)|$): $|P| - |P \cap C| - |P \cap J| + |P \cap C \cap J| = 105 - 50 - 15 + 15 = 55$

• Candidates proficient only in C ($|C \setminus (P \cup J)|$): $|C| - |C \cap P| - |C \cap J| + |C \cap P \cap J| = 95 - 50 - 25 + 15 = 35$

• Candidates proficient only in Java ($|J \setminus (P \cup C)|$): $|J| - |P \cap J| - |C \cap J| + |P \cap C \cap J| = 115 - 15 - 25 + 15 = 90$

Add these numbers to find the total number of candidates proficient in exactly one programming language: $55 + 35 + 90 = 180$

Thus, the number of candidates who are proficient in exactly one of the three programming languages is 180.

Do you have any questions or need further details?

Here are 8 related questions to further explore the topic:

1. How many candidates are proficient in at least two programming languages?
2. What is the total number of candidates proficient in Python but not in C or Java?
3. How many candidates are proficient in all three programming languages?
4. If 5 additional candidates were proficient in Python and Java, how would this affect the total number of candidates knowing exactly one language?
5. What is the probability that a randomly chosen candidate is proficient in C?
6. How many candidates are proficient in Python and C, but not in Java?
7. If the number of candidates who don’t know any programming languages decreased by 5, how would this impact the analysis?
8. How can Venn diagrams be used to visually represent this data?

Tip: Understanding the principle of inclusion-exclusion is crucial for solving problems involving overlapping sets and can be applied to various fields including probability, statistics, and database query optimization.