This year, on which day does Rishi's birthday fall? Statements I. His birthday…

Concept: In Data Sufficiency questions, judge each statement's power to fix a single, unique answer on its own first; only when neither statement alone succeeds do you check whether combining them removes every remaining ambiguity. A calendar clue that names a date range together with one date's weekday fixes the weekday of every date in that stretch; a clue that only excludes one weekday out of seven, with no date-range information of its own, cannot by itself identify which specific date the birthday falls on.

1. Statement I fixes June 27 as a Tuesday and places the birthday strictly inside the window running from June 27 to June 30 — that leaves June 28 (Wednesday) and June 29 (Thursday) as the two dates consistent with it, so Statement I alone cannot choose between them.

2. Statement II states only that the birthday is not on a Thursday — this excludes just one weekday out of seven, and with no month or date-range attached, it cannot by itself identify any specific date, so Statement II alone is nowhere near sufficient.

3. Combining them: apply Statement II's day-of-week exclusion to the two dates that Statement I already isolated. June 29 falls on Thursday, so it is ruled out; June 28 falls on Wednesday, so it survives and is the only date left.

Cross-check: Checking the surviving date against both clues confirms it: it sits inside the June 27–30 window implied by Statement I, and its weekday is not Thursday, so it satisfies Statement II as well — no other date inside that window does.

Neither statement decides the birthday alone, but together they leave exactly one date standing, so both statements put together are sufficient.