The correct method is (3) Treatment with 5‑bromo‑2′-deoxyuridine (BrdU), followed by UV‑A exposure.

Why option (3) works

BrdU (5‑bromo‑2′-deoxyuridine) is a thymidine analog that is incorporated into DNA only in cells undergoing S phase.
When these BrdU-labelled cells are exposed to UV‑A light, the brominated bases make DNA highly photosensitive, leading to strand breaks and lethal DNA damage specifically in the BrdU‑positive (i.e., newly dividing) cells.​
Non-dividing cells, which did not incorporate BrdU, experience much less damage and thus survive, giving selective killing of proliferating cells.

Why the other options are not appropriate

(1) UV radiation at 250 nm

• UV‑C (~250–260 nm) causes strong DNA damage (pyrimidine dimers) in all exposed cells, irrespective of whether they are dividing.

• It is not selective for newly dividing cells and will kill quiescent cells as well.​

(2) EdU followed by doxorubicin

• EdU (5‑ethynyl‑2′-deoxyuridine) is also a thymidine analog used mainly for click‑chemistry based S‑phase detection, not for targeted killing.

• Doxorubicin is a general chemotherapeutic that intercalates DNA and inhibits topoisomerase II, affecting many cells, not only those that recently incorporated EdU.

• This combination does not give a well‑defined, BrdU/UV‑like selective ablation of dividing cells.​

(4) Tritiated thymidine followed by vinblastine

• [³H]‑thymidine can be used for “suicide” labeling of proliferating cells, but it is primarily a radioactive tracer; selective killing requires high doses and long times and is less controlled.

• Vinblastine disrupts microtubules and arrests cells in mitosis but acts on all cells entering mitosis, not only those recently labeled; together this pair is not the standard, precise method for selective ablation of newly dividing cells.​

Using BrdU incorporation followed by UV‑A exposure (option 3) is therefore the most appropriate and widely used method for selectively killing newly dividing mammalian cells in culture.