How many of the cubes have at least 2 faces painted?

Let us see the changes due to removal of cube from corner-
Number of vertices with three faces exposed (Painted) is 7 + 3 = 10
Number of Cubes with 2 sides exposed (Painted): In general one edge gives us 4 (n - 2 in general case) cubes with two face painted but in this case out of 12 edges only 9 edges will give us 4 cubes in one edge and remaining 3 edges will give us 3 cubes from one edge, hence total number of edge is 9 x 4 + 3 x 3 = 45
Number of Cubes with 1 side exposed (Painted): It will remain same as normal case i.e. 6(4²) = 96
Number of Cubes with no sides exposed (Painted) is 4³ = 64
From the above observation:
From the above explanation number of the cubes with at least 2 faces painted is 45 + 10 = 55.