To solve this, let's break it down step by step.

We are given a cuboid with dimensions $5 \times 6 \times 7$, and it has been cut into unit cubes. The total number of unit cubes in the cuboid is:

$5 \times 6 \times 7 = 210$

Now, we need to determine how many of these unit cubes have 0 faces painted. These are the cubes that are completely inside the cuboid, meaning they do not lie on the surface.

To find this, we exclude the outermost layer from all sides. The dimensions of the inner cuboid are obtained by subtracting 2 from each dimension (since both sides of each dimension are painted):

• In the length dimension: $5 - 2 = 3$
• In the width dimension: $6 - 2 = 4$
• In the height dimension: $7 - 2 = 5$

Thus, the inner cuboid (which contains the unit cubes with 0 faces painted) has dimensions $3 \times 4 \times 5$, and the number of unit cubes inside it is:

$3 \times 4 \times 5 = 60$

So, the number of unit cubes with 0 faces painted is 60.

Would you like further details or have any questions?

Here are 5 related questions you might explore:

1. How many unit cubes have exactly 1 face painted?
2. How many unit cubes have exactly 2 faces painted?
3. How many unit cubes have exactly 3 faces painted?
4. What is the total surface area of the original cuboid?
5. If the cuboid were larger, say $8 \times 9 \times 10$, how would the calculations change?

Tip: To find cubes with painted faces, consider the boundaries and corners, where cubes with more painted faces are located!