Fill in the blank using the most suitable connector from the options…

The correct answer is Option (3) → as

Conjunction of reason:

• As, because and since introduce the reason for what the main clause says.
• Yet, still and but show contrast: the second idea goes against what the first leads us to expect.
• Deciding clue: "it was bound to be rejected" gives the reason why the advice was futile.
• This sentence: the advice was futile (useless) because it was bound to be rejected, so the blank needs a word of reason, as.
• Complete sentence: I knew giving advice to him was futile as it was bound to be rejected.
  Example: We stayed indoors as it was raining heavily.

Why the Other Options are Incorrect?

Option 1: yet — Incorrect. Yet joins two ideas that contrast; here the second clause is the cause of the first, not a contrast.
Example: He is rich, yet he is unhappy.

Option 2: still — Incorrect. Still means even so, in spite of what was said; it does not give a reason.
Example: It was raining; still, we went out.

Option 4: but — Incorrect. But joins two contrasting ideas; it does not give a reason.
Example: She tried hard but failed.

Contrast: as gives a reason; "yet", "still" and "but" show contrast.