Question: A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three colors, red, blue, and green are used to paint the six faces of the cube. If the adjacent faces are painted with the different colors, in how many ways can the cube be painted?

1. 3
2. 6
3. 8
4. 12
5. 27

Answer:
Approach Solution (1):
If the base of the cube is red, then in order the adjacent faces to be painted with the different colors, the top must also be red. 4 side faces can be painted in Green-Blue-Green-Blue or Blue-Green-Blue-green (2 options)
But we have the base painted in either of the three colors, thus the total number of ways to paint the cube is 3 * 2 = 6 ways
Correct option: B

Approach Solution (2):
Let’s say side 1 is painted red, and then the 4 adjacent sides can be either green or blue alternating. This can be done in 2 ways:
GBGB
BGBG
Sixth side should be the same color as side 1
For each color chosen for side 1 (and side 6), there are 2 ways of paining the side 2, 3, 4, and 5.
Number of colors that can be chosen for side 1 (and side 6) is 3
So, 3 * 2 = 6
Correct option: B

Approach Solution (3):
Imagine a flattened cube. The three colored region will establish the other colors of the remaining faces of the cube
For example: We assumed the sequence of color in the given image as Red on face #1 and Blue on the face #2 and Green on the face #3. Since face 1 Is red then we know 4 and 5 cannot be red. Since the face 2 is blue, we know that 5 and 6 cannot be blue. Since the
face 3 is green, we know 4 and 6 (the bottom) cannot be green
So, all we need is to count the possible number of arrangements of 3 colors.
3! = 6
Correct option: B

“A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three colors, red, blue, and green are used to paint the six faces of the cube. If the adjacent faces are painted with the different colors, in how many ways can the cube be painted?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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