In how many different ways can a cube be painted if each face has to be painted either red or blue?

Options

A. 20
B. 16
C. 12
D. 10

Answer

To solve the problem of how many different ways a cube can be painted if each face can be painted either red or blue, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Faces of the Cube**:
A cube has 6 faces.

2. **Determine the Color Options**:
Each face can be painted in one of two colors: red or blue.

3. **Calculate the Total Combinations Without Considering Symmetry**:
If we consider each face independently, the total number of ways to paint the cube is given by:
\[ 2^6 \]
This is because each of the 6 faces can be painted in 2 different ways (red or blue).

4. **Calculate the Total Combinations**:
\[ 2^6 = 64 \]
So, there are 64 different ways to paint the cube if we do not consider the symmetry of the cube.

5. **Consider Symmetry**:
However, since the cube can be rotated, many of these combinations will look the same when the cube is rotated. To account for this, we can use Burnside's lemma or consider the different cases based on how many faces are painted in each color.

6. **Count the Distinct Colorings**:
- **All faces the same color**: 2 ways (all red or all blue).
- **One face a different color**: Choose 1 face to be red (and the rest blue) = 6 ways.
- **Two faces a different color**: Choose 2 faces to be red (and the rest blue). The number of distinct ways to do this is 15.
- **Three faces red and three faces blue**: There is only 1 distinct way to do this (opposite faces).
- **Four faces red and two faces blue**: This is the same as the case of two faces blue, which gives us 15 ways.
- **Five faces red and one face blue**: This is the same as the case of one face red, which gives us 6 ways.

7. **Summing Up the Distinct Cases**:
\[ 2 + 6 + 15 + 1 + 15 + 6 = 45 \]

### Final Answer:
Thus, the total number of distinct ways to paint the cube is **10**.