Probability
Classical Probability
Grade 12
Question:
<p><strong>For Problems 7–9</strong><br>A cube having all of its sides painted is cut by two horizontal, two vertical, and other two planes so as to form 27 cubes all having the same dimensions. Of these cubes, a cube is selected at random.</p><p><strong>Problem 9:</strong> The total number of cubes having at least one of its sides painted is</p>
<p>18</p>
<p>20</p>
<p>22</p>
<p>26</p>
Step-by-Step Solution
Key Concept: A 3×3×3 cube divided into 27 unit cubes has 8 corner cubes (3 faces painted), 12 edge cubes (2 faces painted), 6 face cubes (1 face painted), and 1 center cube (0 faces painted). Cubes with 'at least one face painted' = Total - Unpainted = 27 - 1 = 26.
<p><strong>Step 1:</strong> When a 3×3×3 cube is cut by planes parallel to faces, it creates 27 unit cubes arranged in a 3×3×3 grid.</p><p><strong>Step 2:</strong> Classify cubes by painted faces:</p><ul><li><strong>Corner cubes:</strong> 8 cubes with 3 faces painted</li><li><strong>Edge-center cubes:</strong> 12 cubes with 2 faces painted</li><li><strong>Face-center cubes:</strong> 6 cubes with 1 face painted</li><li><strong>Center cube:</strong> 1 cube with 0 faces painted</li></ul><p><strong>Step 3:</strong> Cubes with at least one face painted = 8 + 12 + 6 = 26</p><p><strong>Verification:</strong> Total cubes - Unpainted cubes = 27 - 1 = 26 ✓</p><p>∴ Answer: <strong>26</strong></p>
Correct Answer: D