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DIRECTIONS for the question: Read the information given below and answer the question that follows.
Four colors-White, Blue, Green and Orange- are used to paint a cube such that each face is painted in exactly one color and each color is painted on at least one face. The cube is now perfectly and completely cut into exactly 120 identical cuboids by making the least possible number of cuts.
Four colors-White, Blue, Green and Orange- are used to paint a cube such that each face is painted in exactly one color and each color is painted on at least one face. The cube is now perfectly and completely cut into exactly 120 identical cuboids by making the least possible number of cuts.
Q. What is the least possible number of cuboids which have at most one color on them? (in numerical value)
Correct answer is '76'. Can you explain this answer?
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DIRECTIONSfor the question:Read the information given below and answer...
Here we have to consider the least possible number of cuts needed for getting 120 pieces.
120 = 4 ×5 × 6 ⇒ (4 - 1) + (5 - 1) + (6 - 1) = 12
As shown in the above figure , any face can have 30 or 24 or 20 cuboids. Any edge can have 6 or 5 or 4 cuboids.
To get least possible number of such cuboids all the cuboids on the edges (and corners) must have more than one color which is possible when opposite faces are painted in the same color. (i.e two pairs of opposite faces )
Number of cuboids with three colors on them = 8
Number of cuboids with two colors on them
= 4 × (2 + 3 + 4) = 36
= 4 × (2 + 3 + 4) = 36
∴ Number of required cuboids (i.e. with exactly one color or no color on them) = 120 - { 36 + 8 } = 76.
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DIRECTIONSfor the question:Read the information given below and answer...
Given Information:
- Four colors (White, Blue, Green, and Orange) are used to paint a cube.
- Each face of the cube is painted in exactly one color.
- Each color is painted on at least one face.
- The cube is cut into 120 identical cuboids.
Analysis:
To find the least possible number of cuboids with at most one color, we need to determine the arrangement of colors on the cube and how the cube is cut.
Arrangement of Colors on the Cube:
Since each face of the cube is painted in exactly one color and each color is painted on at least one face, there are two possible arrangements:
1. Two opposite faces have the same color, while the other four faces have different colors.
2. Each face has a different color.
Arrangement 1: Two opposite faces have the same color:
In this arrangement, there are 4 possible ways to choose the color for the two opposite faces. Once the two opposite faces are chosen, the other four faces can be assigned colors in 3! (3 factorial) ways. Therefore, there are a total of 4 x 3! = 24 different arrangements for this case.
Arrangement 2: Each face has a different color:
In this arrangement, there are 4! (4 factorial) ways to assign colors to the four faces of the cube.
Number of Cuboids:
To find the least possible number of cuboids, we need to determine the minimum number of cuts required to divide the cube into 120 identical cuboids.
Since each cut creates 2 new faces, the number of cuts required to create 120 identical cuboids is equal to the total number of faces on the original cube minus 1. Therefore, the number of cuts required is 6 - 1 = 5.
Each cut can intersect at most 2 colors. Therefore, the number of cuboids with at most one color is equal to the number of colors (4) multiplied by the number of cuts (5).
Calculation:
Number of cuboids with at most one color = Number of colors x Number of cuts
= 4 x 5
= 20
However, we need to consider both arrangements of colors on the cube. Therefore, the total number of cuboids with at most one color is 20 x 2 = 40.
Conclusion:
The least possible number of cuboids with at most one color is 40, not 76 as mentioned in the question.
- Four colors (White, Blue, Green, and Orange) are used to paint a cube.
- Each face of the cube is painted in exactly one color.
- Each color is painted on at least one face.
- The cube is cut into 120 identical cuboids.
Analysis:
To find the least possible number of cuboids with at most one color, we need to determine the arrangement of colors on the cube and how the cube is cut.
Arrangement of Colors on the Cube:
Since each face of the cube is painted in exactly one color and each color is painted on at least one face, there are two possible arrangements:
1. Two opposite faces have the same color, while the other four faces have different colors.
2. Each face has a different color.
Arrangement 1: Two opposite faces have the same color:
In this arrangement, there are 4 possible ways to choose the color for the two opposite faces. Once the two opposite faces are chosen, the other four faces can be assigned colors in 3! (3 factorial) ways. Therefore, there are a total of 4 x 3! = 24 different arrangements for this case.
Arrangement 2: Each face has a different color:
In this arrangement, there are 4! (4 factorial) ways to assign colors to the four faces of the cube.
Number of Cuboids:
To find the least possible number of cuboids, we need to determine the minimum number of cuts required to divide the cube into 120 identical cuboids.
Since each cut creates 2 new faces, the number of cuts required to create 120 identical cuboids is equal to the total number of faces on the original cube minus 1. Therefore, the number of cuts required is 6 - 1 = 5.
Each cut can intersect at most 2 colors. Therefore, the number of cuboids with at most one color is equal to the number of colors (4) multiplied by the number of cuts (5).
Calculation:
Number of cuboids with at most one color = Number of colors x Number of cuts
= 4 x 5
= 20
However, we need to consider both arrangements of colors on the cube. Therefore, the total number of cuboids with at most one color is 20 x 2 = 40.
Conclusion:
The least possible number of cuboids with at most one color is 40, not 76 as mentioned in the question.
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DIRECTIONSfor the question:Read the information given below and answer the question that follows.Four colors-White, Blue, Green and Orange- are used to paint a cube such that each face is painted in exactly one color and each color is painted on at least one face. The cube is now perfectly and completely cut into exactly 120 identical cuboids by making the least possible number of cuts.Q. What is the least possible number of cuboids which have at most one color on them? (in numerical value)Correct answer is '76'. Can you explain this answer?
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DIRECTIONSfor the question:Read the information given below and answer the question that follows.Four colors-White, Blue, Green and Orange- are used to paint a cube such that each face is painted in exactly one color and each color is painted on at least one face. The cube is now perfectly and completely cut into exactly 120 identical cuboids by making the least possible number of cuts.Q. What is the least possible number of cuboids which have at most one color on them? (in numerical value)Correct answer is '76'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about DIRECTIONSfor the question:Read the information given below and answer the question that follows.Four colors-White, Blue, Green and Orange- are used to paint a cube such that each face is painted in exactly one color and each color is painted on at least one face. The cube is now perfectly and completely cut into exactly 120 identical cuboids by making the least possible number of cuts.Q. What is the least possible number of cuboids which have at most one color on them? (in numerical value)Correct answer is '76'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for DIRECTIONSfor the question:Read the information given below and answer the question that follows.Four colors-White, Blue, Green and Orange- are used to paint a cube such that each face is painted in exactly one color and each color is painted on at least one face. The cube is now perfectly and completely cut into exactly 120 identical cuboids by making the least possible number of cuts.Q. What is the least possible number of cuboids which have at most one color on them? (in numerical value)Correct answer is '76'. Can you explain this answer?.
DIRECTIONSfor the question:Read the information given below and answer the question that follows.Four colors-White, Blue, Green and Orange- are used to paint a cube such that each face is painted in exactly one color and each color is painted on at least one face. The cube is now perfectly and completely cut into exactly 120 identical cuboids by making the least possible number of cuts.Q. What is the least possible number of cuboids which have at most one color on them? (in numerical value)Correct answer is '76'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about DIRECTIONSfor the question:Read the information given below and answer the question that follows.Four colors-White, Blue, Green and Orange- are used to paint a cube such that each face is painted in exactly one color and each color is painted on at least one face. The cube is now perfectly and completely cut into exactly 120 identical cuboids by making the least possible number of cuts.Q. What is the least possible number of cuboids which have at most one color on them? (in numerical value)Correct answer is '76'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for DIRECTIONSfor the question:Read the information given below and answer the question that follows.Four colors-White, Blue, Green and Orange- are used to paint a cube such that each face is painted in exactly one color and each color is painted on at least one face. The cube is now perfectly and completely cut into exactly 120 identical cuboids by making the least possible number of cuts.Q. What is the least possible number of cuboids which have at most one color on them? (in numerical value)Correct answer is '76'. Can you explain this answer?.
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