Select the appropriate word to fill in the blank:Rahul thought he had…

The correct answer is Option (3) → since

Conjunction of reason:

• Since (like "because" and "as") introduces the reason for what is said in the main clause.
• This sentence: Rahul thought he had a good chance because his uncle was on the company's board of directors; the second clause gives the reason, so "since" joins the two.
• Complete sentence: Rahul thought he had a good chance to get the job since his uncle was on the company's board of directors.
  Example: Since it was raining, the match was put off.

Why the Other Options are Incorrect?

Option 1: but — Incorrect. But shows contrast; the uncle's position supports Rahul's hope, it does not go against it.
Example: He tried hard, but he could not win.

Option 2: yet — Incorrect. Yet shows an unexpected contrast (in spite of that); there is no contrast between the two clauses.
Example: She was tired, yet she kept working.

Option 4: unless — Incorrect. Unless means "if not" and states a condition; "a good chance unless his uncle was on the board" makes no sense.
Example: You cannot enter unless you have a pass.

Contrast: "since" gives a reason, "but" and "yet" show a contrast, and "unless" sets a condition.