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Difference between revisions of "2023 AIME I Problems/Problem 3"
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Unofficial problem statement:
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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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In a plane there exists <math>40</math> lines, none of which are parallel. There are <math>3</math> points in which exactly <math>3</math> lines intersect, <math>4</math> points in which exactly <math>4</math> lines intersect, <math>5</math> points in which exactly <math>5</math> lines intersect, and <math>6</math> points in which exactly <math>6</math> lines intersect. There are no points in which more than <math>6</math> lines intersect. How many points are there in which exactly <math>2</math> lines intersect?
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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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exactly 2 lines intersect.
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==Solution==
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The number of points where
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exactly 2 lines intersect is
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<cmath>
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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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- 5 \cdot \binom{5}{2} - 6 \cdot \binom{6}{2}  \\
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& = \boxed{\textbf{(607) }} .
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\end{align*}
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</cmath>
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~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Revision as of 14:37, 8 February 2023

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

The number of points where exactly 2 lines intersect is \begin{align*} & \binom{40}{2} - 3 \cdot \binom{3}{2} - 4 \cdot \binom{4}{2} - 5 \cdot \binom{5}{2} - 6 \cdot \binom{6}{2} \\ & = \boxed{\textbf{(607) }} . \end{align*}

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

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