# A problem in elementary combinatorial space geometry.

Consider $3$ dimensional projective space (although you don't have to know anything specific, just that there is no parallelism.) Here is the question:

How many pairs of intersecting lines in space are there such that:

1) Their plane passes through a given fixed line $l$.

2) Their point of intersection lies on a given plane $P$.

3) Together they intersect $4$ given lines $l_1,l_2,l_3,l_4$ ?

The answer in the book listed is $17$, assuming there is no typo. The following is useful.

$\bf{Theorem}$ Given four lines in space, there are $2$ lines intersecting all four.

${\bf Thoughts :}$ Well, you don't have to read this if you want to approach the question fresh, but I can tell you where my problem lies, so you get the core quicker.

First, either one of the lines intersects $3$ of $l_1,l_2,l_3,l_4$, or each line intersects $2$ of them.

In the first case there are $\binom{4}{3}=4$ many triples of lines from $l_1,l_2,l_3,l_4$, say $l_1,l_2,l_3$, and together with $l$ there are two lines intersecting all four. Each of these intersects $P$ is a point. Now all that is needed is a line through this point and the lines $l$ and $l_4$, this is unique as is easy to see. This gives $2\times 4=8$ line pairs.

In the second case each line must intersect $2$ of the lines $l_1,l_2,l_3,l_4$. There are $\frac{1}{2}\binom{4}{2}=3$ many ways to divide this. Now if the book is right for each such division there must be $3$ solutions. This is were I cannot proceed further.

• Now there is a related question on MathOverflow: Some Elementary Schubert Calculus Calculations. Mar 30, 2018 at 2:28
• @MartinSleziak Yes, I posted it, since there seemed no interest in the question here. Mar 30, 2018 at 2:56
• Yes I've added a link because it is recommended that each post should be linked to other copies if a question is cross-posted, see meta: Moderator Supported (Official) Guidelines for “Legitimate” CrossPosting?. (The questions are not exactly the same, but still I thought that adding the link might be useful.) Mar 30, 2018 at 4:52
• @MartinSleziak Yeah, the other answer has received an answer, I was thinking of deleting this one. Dont know if that is bad form or not. Mar 30, 2018 at 12:45
• There is a post on meta: What to do with my cross-posted question that got answered elsewhere? The suggestion there is to self-answer the question if it got satisfactory answer on another site. This might also earn you self-learner badge :-) Apr 2, 2018 at 10:29