Explanation:

- Start by calculating the volume of the large cube:

- The cube's edge is 48 cm, so the volume is $48\times 48\times 48=110592$ cm${}^3$.

- Next, calculate how many small cubes can be made:

- Each small cube has a volume of 64 cm${}^3$.

- Divide the large cube's volume by a small cube's volume to find the number of small cubes: $\frac{110592}{64}=1728$.

- Therefore, option 1: 1728 small cubes can be made.

- The calculations confirm:

- Option 2: 1718, not correct.

- Option 3: 1748, not correct.

- Option 4: 1738, not correct.

- Hence, the correct answer is Option 1: 1728.