Permutations & Combinations
Plane Division by Lines
Grade 11
Question:
<p>There are <i>n</i> lines in a plane, no two of which are parallel and no three of them are concurrent. Let the plane be divided by <i>n</i> lines in <i>a<sub>n</sub></i> parts, then</p>
<p>(A) <i>a<sub>n</sub></i> = <i>a</i><sub><i>n</i>–1</sub> + (<i>n</i> – 1)</p>
<p>(B) <i>a<sub>n</sub></i> = <i>a</i><sub><i>n</i>–1</sub> + <i>n</i></p>
<p>(C) <i>a</i><sub>6</sub> = 22</p>
<p>(D) <i>a</i><sub>10</sub> = 56</p>
Step-by-Step Solution
Key Concept: Each new line intersects all previous lines at distinct points, adding exactly <i>n</i> new regions.
<p><strong>Analysis:</strong> When the <i>n</i>-th line is added to <i>n</i>–1 existing lines, it intersects each at a distinct point (no three concurrent), creating <i>n</i> new regions. Thus the recurrence is:</p><p>\[a_n = a_{n-1} + n\]</p><p>With <i>a</i><sub>1</sub> = 2, we get: <i>a<sub>n</sub></i> = 1 + 1 + 2 + 3 + ... + <i>n</i> = 1 + \(\frac{n(n+1)}{2}\)</p><p>For <i>n</i> = 6: <i>a</i><sub>6</sub> = 1 + 21 = 22 ✓</p><p>For <i>n</i> = 10: <i>a</i><sub>10</sub> = 1 + 55 = 56 ✓</p>
Correct Answer: B