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Given a diagram as follows.

enter image description here

Only dots on AD are collinear, so are those on BC. The objective is to find the numbers of lines passing through the dots. Note that any overlapped lines are considered as a single line.

My attempt:

  • No collinear dots (on AB and CD) contribute $C_2^4=6$ lines.
  • Every collinear dot on BC can be paired with a single dot on CD. It contributes $3\times 2=6$ lines.
  • Every collinear dot on BC can be paired with a single dot on AB. It contributes $3\times 2=6$ lines.
  • Every collinear dot on AD can be paired with a single dot on CD. It contributes $4\times 2=8$ lines.
  • Every collinear dot on AD can be paired with a single dot on AB. It contributes $4\times 2=8$ lines.
  • A single dot on AD can paired with a single dot on BC. It contributes $4\times 3=12$ lines.
  • There is a line passing through dots on BC.
  • There is a line passing through dots on AD.

There are 48 lines in total.

Question

Is there any simpler method?

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  • $\begingroup$ Using generating functions is possible? $\endgroup$ Aug 11 '20 at 9:47
  • $\begingroup$ Do the dots have dimensions? I see different sizes.... $\endgroup$
    – dmtri
    Aug 11 '20 at 9:56
  • $\begingroup$ @dmtri: No. They have no dimension. $\endgroup$ Aug 11 '20 at 9:57
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A simpler approach would be to count all pairs of dots, and then subtract multiply counted lines.

If no two dots were collinear, we'd have ${11 \choose 2} = 55$ lines.

Now, this counts the line $AD$ ${4 \choose 2} = 6$ times, so we need to subtract 5.

Also, we've counted the line $BC$ ${3 \choose 2} = 3$ times, so we need to subtract 2.

Overall, we have ${11 \choose 2} - ({4 \choose 2} - 1) - ({3 \choose 2} - 1) = 55 - 5 - 2 = 48$ lines.

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If we take any pair of points we get $${11\choose2}= 55$$ lines: But some of them are counted mor than once. The one in on upper segment aand lower segment. So we have to substract ${4\choose 2}-1= 5$ and ${3\choose 2}-1 = 2$ lines.

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  • $\begingroup$ OK. Thanks. It is the same as the previous answer. $\endgroup$ Aug 11 '20 at 10:04
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    $\begingroup$ Yes, I started to write down and didn't read it before posting mine. $\endgroup$
    – Aqua
    Aug 11 '20 at 10:07
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If you take one of the points, it will have 10 lines, then the next will have additional 9 and so on,

So total number of lines = $\dfrac {10(10+1)}{2} = 55$

There are $6$ lines on the top horizontal length and $3$ lines on the bottom horizontal length. Only $2$ of them are unique. So we should subtract $7$ from $55$.

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