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Difference between revisions of "2023 AIME I Problems/Problem 3"
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A plane contains 40 lines, no 2 of which are parallel. Suppose that there are 3 points where exactly 3 lines
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==Problem==
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intersect, 4 points where exactly 4 lines intersect, 5 points where exactly 5 lines intersect, 6 points where
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exactly 6 lines intersect, and no points where more than 6 lines intersect. Find the number of points where
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A plane contains <math>40</math> lines, no <math>2</math> of which are parallel. Suppose that there are <math>3</math> points where exactly <math>3</math> lines intersect, <math>4</math> points where exactly <math>4</math> lines intersect, <math>5</math> points where exactly <math>5</math> lines intersect, <math>6</math> points where exactly <math>6</math> lines intersect, and no points where more than <math>6</math> lines intersect. Find the number of points where exactly <math>2</math> lines intersect.
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exactly 2 lines intersect.
 
  
 
==Solution==
 
==Solution==
  
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The number of points where
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In this solution, let <b><math>\boldsymbol{n}</math>-line points</b> be the points where exactly <math>n</math> lines intersect. We wish to find the number of <math>2</math>-line points.
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exactly 2 lines intersect is
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<cmath>
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There are <math>\binom{40}{2}=780</math> pairs of lines. Among them:
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\begin{align*}
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& \binom{40}{2} - 3 \cdot \binom{3}{2} - 4 \cdot \binom{4}{2}
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* The <math>3</math>-line points account for <math>3\cdot\binom32=9</math> pairs of lines.
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- 5 \cdot \binom{5}{2} - 6 \cdot \binom{6}{2}  \\
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& = \boxed{\textbf{(607) }} .
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* The <math>4</math>-line points account for <math>4\cdot\binom42=24</math> pairs of lines.
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\end{align*}
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</cmath>
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* The <math>5</math>-line points account for <math>5\cdot\binom52=50</math> pairs of lines.
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* The <math>6</math>-line points account for <math>6\cdot\binom62=90</math> pairs of lines.
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It follows that the <math>2</math>-line points account for <math>780-9-24-50-90=607</math> pairs of lines.
  
 
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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~MRENTHUSIASM
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==See also==
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{{AIME box|year=2023|num-b=2|num-a=4|n=I}}

Revision as of 14:17, 8 February 2023

Problem

A plane contains $40$ lines, no $2$ of which are parallel. Suppose that there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect.

Solution

In this solution, let $\boldsymbol{n}$-line points be the points where exactly $n$ lines intersect. We wish to find the number of $2$-line points.

There are $\binom{40}{2}=780$ pairs of lines. Among them:

  • The $3$-line points account for $3\cdot\binom32=9$ pairs of lines.
  • The $4$-line points account for $4\cdot\binom42=24$ pairs of lines.
  • The $5$-line points account for $5\cdot\binom52=50$ pairs of lines.
  • The $6$-line points account for $6\cdot\binom62=90$ pairs of lines.

It follows that the $2$-line points account for $780-9-24-50-90=607$ pairs of lines.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

~MRENTHUSIASM

See also

2023 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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