Hint: Think about one face at a time.

Answer: 1,352 small cubes.

The original large cube with 16 small cubes along each edge would required 16 x 16 x 16 = 4,096 small cubes.

Removing the outside layer would leave a large cube now with 14 small cubes along each edge (which requires 14 x 14 x 14 = 2,744 small cubes).

It isn't 15 small cubes because you're removing both ends of an edge.

Therefore you have removed 4,096 - 2,744 = 1,352 small cubes.