To arrange the given materials—Germanium, Silicon, Gallium Arsenide, and Gallium Nitride—in ascending order based on their electron mobility at 300 K, we need to understand that electron mobility is a critical property which indicates how quickly an electron can move through a semiconductor material when subjected to an electric field.

Electron mobility is influenced by the material's crystalline structure, atomic bonding, and purity. Typically, compound semiconductors like Gallium Arsenide have higher electron mobility compared to elemental semiconductors like Silicon and Germanium.

1. Germanium: It is a group IV semiconductor with relatively high electron mobility due to its crystalline structure but generally lower than compound semiconductors like Gallium Arsenide.
2. Silicon: Another group IV semiconductor, Silicon has lower electron mobility than Germanium. It is widely used in the electronics industry due to its abundance and stable properties.
3. Gallium Arsenide: A compound semiconductor with higher electron mobility than either Silicon or Germanium, due to its direct bandgap and superior electron transport properties.
4. Gallium Nitride: While known for high-power and high-frequency applications, its electron mobility is generally less than Gallium Arsenide but higher than Silicon and Germanium due to its wide bandgap characteristics.

Based on these observations, we can list the materials in increasing order of their electron mobility as follows:

1. Silicon
2. Germanium
3. Gallium Nitride
4. Gallium Arsenide

Thus, the correct option that arranges these materials in ascending order of their electron mobility is: (d) (b) (a) (c)