Mrs. Jones had some white paint and some green paint, and a bunch of wooden cubes. Her class decided to paint the cubes by making each face either solid white or green. Juan painted his cube with all 6 faces white - Julie painted her cube solid green. Hector painted 4 faces white and 2 faces green. How many cubes could be painted in the fashion, so that each cube is different from the others? Two cubes are alike if one can be turned so that is exactly matches, color for color on each side, the other cube.

The all white or all green can only be one way. The two green-faced sides could either be opposite or adjacent.

Answer = 4

To find how many cubes can be painted differently, we need to consider the number of different combinations of colors for each face.

Let's break it down step by step:

First, we consider Juan's cube, which is painted white on all six faces. In this case, there is only one cube that can be painted like Juan's since all the faces have the same color.

Next, we consider Julie's cube, which is painted green on all six faces. Like Juan's cube, there is also only one cube that can be painted like Julie's because all faces have the same color.

Moving on to Hector's cube, which has 4 white faces and 2 green faces. To determine how many cubes can be painted like Hector's, we need to calculate the combinations of choosing 4 out of the 6 faces to be white. This can be done using binomial coefficients, also known as "n choose k" in combinatorics.

The binomial coefficient formula is given by: C(n, k) = n! / (k! * (n-k)!)

In this case, n = 6 (total number of faces) and k = 4 (number of white faces). Plugging these values into the formula, we get:

C(6, 4) = 6! / (4! * (6-4)!)
= 6! / (4! * 2!)
= (6 * 5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (2 * 1))
= 15

Therefore, there are 15 different cubes that can be painted like Hector's cube.

In conclusion, there are a total of 1 + 1 + 15 = 17 different cubes that can be painted in the fashion described, meaning each cube is different from the others.