Step 1: Understanding the sentence structure.
The sentence provided has a contrast between the writer's happiness at getting into IIT and the fact that his joy was short-lived. The word “though” is used to introduce a contrast, but it should not be followed by “but” in the same sentence.
Step 2: Analyzing the options.
(1) Though I was quite happy getting into IIT, but my joy was short-lived: Incorrect — The conjunction "though" already introduces the contrast, so the word "but" is redundant.
(2) I was quite happy getting into IIT, though my joy is short-lived: Incorrect — The tense of "is" is not consistent with the past tense of "was."
(3) Though I was quite happy getting into IIT, my joy was short-lived: Correct — This option properly uses "though" to introduce the contrast and maintains consistent tense.
(4) I was quite happy though getting into IIT, my joy was short-lived: Incorrect — The word "though" is incorrectly placed, as it should follow the entire phrase of happiness, not be in the middle.
Step 3: Conclusion.
The correct answer is (3) Though I was quite happy getting into IIT, my joy was short-lived, as it maintains proper sentence structure and tense agreement.