To find the number of candidates proficient in exactly one of the three programming languages (Python, C, and Java), we will use the principle of inclusion-exclusion.

Given Data

• Total candidates ($N$) = 500
• Candidates proficient in:
  • Python ($|P|$) = 285
  • C ($|C|$) = 195
  • Java ($|J|$) = 115
• Candidates proficient in:
  • Python and Java ($|P \cap J|$) = 45
  • C and Python ($|C \cap P|$) = 70
  • C and Java ($|C \cap J|$) = 50
• Candidates proficient in none of the languages = 50

Let’s denote the following sets:

• $P$: Candidates proficient in Python
• $C$: Candidates proficient in C
• $J$: Candidates proficient in Java

Step 1: Find Total Candidates Proficient in At Least One Language

Since 50 candidates don’t know any language, the number of candidates proficient in at least one language is: $500 - 50 = 450$

Step 2: Use Inclusion-Exclusion Principle

The formula for the total number of candidates proficient in at least one of the three languages is: $|P \cup C \cup J| = |P| + |C| + |J| - |P \cap C| - |C \cap J| - |P \cap J| + |P \cap C \cap J|$ We need to find $|P \cap C \cap J|$ (candidates proficient in all three languages).

Rearranging for $|P \cap C \cap J|$: $450 = 285 + 195 + 115 - 70 - 50 - 45 + |P \cap C \cap J|$ $450 = 430 + |P \cap C \cap J|$ $|P \cap C \cap J| = 450 - 430 = 20$

Step 3: Calculate Candidates Proficient in Exactly One Language

To find candidates proficient in only one language, we need to exclude those who know two or three languages.

1. Python only: $|P \text{ only}| = |P| - |P \cap C| - |P \cap J| + |P \cap C \cap J|$ Substituting values: $|P \text{ only}| = 285 - 70 - 45 + 20 = 190$

2. C only: $|C \text{ only}| = |C| - |C \cap P| - |C \cap J| + |P \cap C \cap J|$ Substituting values: $|C \text{ only}| = 195 - 70 - 50 + 20 = 95$

3. Java only: $|J \text{ only}| = |J| - |P \cap J| - |C \cap J| + |P \cap C \cap J|$ Substituting values: $|J \text{ only}| = 115 - 45 - 50 + 20 = 40$

Step 4: Add Up Candidates Proficient in Exactly One Language

$\text{Total proficient in exactly one language} = |P \text{ only}| + |C \text{ only}| + |J \text{ only}|$ $= 190 + 95 + 40 = 325$

Final Answer

The number of candidates proficient in exactly one of the three programming languages is 325.

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Would you like a breakdown of each step? Here are some related questions:

1. What if we add a fourth programming language?
2. How would the solution change if 100 candidates didn’t know any programming language?
3. What if the proficiency in Python, C, and Java had overlapping subgroups?
4. How would adding conditional proficiency (e.g., Java only if C is known) affect the solution?
5. How is the inclusion-exclusion principle used in real-life scenarios?

Tip: Using Venn diagrams is a helpful way to visualize overlapping groups in set theory problems.