The choice of gas mixture for deep diving comes down to two problems that get worse with depth, gases dissolving into the blood under high pressure, and how narcotic a gas is at that pressure. Let's judge each mixture against those two problems.
- Helium - Oxygen: Helium is a very light, chemically inert gas that dissolves far less into fatty tissue and blood than nitrogen does, and it does not produce the narcotic effect nitrogen does under pressure. Pairing it with oxygen for breathing gives divers a mixture that avoids narcosis and cuts the risk of dangerous gas bubbles forming as they surface. This mixture solves both problems.
- Nitrogen - Oxygen: This is just ordinary compressed air. At great depth the nitrogen in it dissolves heavily into the blood and produces a drunken, narcotic state along with a high risk of decompression sickness on ascent, so it fails at depth even though it works fine near the surface.
- Hydrogen - Nitrogen: Hydrogen is extremely flammable and can ignite explosively when mixed with oxygen sources or sparks, and this mixture still carries nitrogen's narcosis risk, so it fails on safety grounds from two directions.
- Helium - Nitrogen: Leaving out oxygen entirely means the diver has nothing to breathe for metabolism, and nitrogen is still present to cause narcosis, so this mixture does not work as a breathing gas at all.
Only the helium-oxygen mixture avoids narcosis, keeps decompression risk manageable, and still supplies the oxygen a diver needs to breathe.
So the correct answer is Helium - Oxygen.