Given below are three statements and four conclusions drawn based on the statements.

Statement $1$ : Some engineers are writers.

Statement $2$ : No writer is an actor.

Statement $3$ : All actors are engineers.

Conclusion $\text{I}$ : Some writers are engineers.

Conclusion $\text{II}$ : All engineers are actors.

Conclusion $\text{III}$ : No actor is a writer.

Conclusion $\text{IV}$ : Some actors are writers.

Which one of the following options can be logically inferred?

1. Only conclusion $\text{I}$ is correct
2. Only conclusion $\text{II}$ and conclusion $\text{III}$ are correct
3. Only conclusion $\text{I}$ and conclusion $\text{III}$ are correct
4. Either conclusion $\text{III}$ or conclusion $\text{IV}$ is correct

Given the statements:

• Statement $1$: Some engineers are writers.
• Statement $2$: No writer is an actor.
• Statement $3$: All actors are engineers.

Based on the above three statements, we can draw the Venn diagram. Based on the above Venn diagram, we can conclude that,

• Conclusion $\text{I:}$ Some writers are engineers.${\color{Green}{-\text{True}}}$
• Conclusion $\text{II:}$ All engineers are actors.${\color{Red}{-\text{False}}}$
• Conclusion $\text{III:}$ No actor is a writer.${\color{Green}{-\text{True}}}$
• Conclusion $\text{IV:}$ Some actors are writers.${\color{Red}{-\text{False}}}$

$\therefore$ Only conclusion $\text{I}$ and conclusion $\text{III}$ are correct.

Correct Answer $:\text{C}$

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