Two cubes of dimension 50cm*50cm*50cm each, are joined by one of the faces. A cuboid of dimension 40cm*30cm*40cm is placed on the top, exactly over the middle of the joined cubes. The entire object is painted and then cut into smaller cubes of dimension 10cm*10cm*10cm. How many no-face painted cubes will be there?

Sure, let's break down the problem step by step.

1. First, let's understand the structure of the object. We have two cubes of dimensions 50cm x 50cm x 50cm each, joined by one of their faces. This forms a larger cuboid of dimensions 100cm x 50cm x 50cm.

2. Then, a cuboid of dimensions 40cm x 30cm x 40cm is placed on top of the joined cubes, exactly in the middle. This means that the total height of the object is now 80cm (50cm from the cubes and 30cm from the cuboid), and the width and length remain 100cm and 50cm respectively.

3. The entire object is then painted and cut into smaller cubes of dimensions 10cm x 10cm x 10cm. This means that the larger object is divided into 10 x 5 x 8 = 400 smaller cubes.

4. The cubes that are painted on no faces are the ones that are completely inside the larger object, i.e., they do not touch any of the outer faces of the larger object. These cubes are in the middle of the larger object, and their number can be calculated by subtracting the outer layer of cubes from the total number of cubes.

5. The outer layer of cubes consists of the cubes that form the top, bottom, and sides of the larger object. The top and bottom layers each have 10 x 5 = 50 cubes, and the four sides each have 8 x 5 = 40 cubes. However, since the corners and edges are counted twice, we need to subtract them out. There are 4 corners on each layer (top and bottom), each consisting of 1 cube, and 4 edges on each layer, each consisting of 8 cubes. So, the total number of cubes in the outer layer is 2*(50 + 440 - 41 - 4*8) = 200 cubes.

6. Therefore, the number of no-face painted cubes is the total number of cubes minus the number of cubes in the outer layer, which is 400 - 200 = 200 cubes.