# Combination of lines

There are $4$ co-linear pints $A_1,A_2,A_3$ and $A_4$ Now suppose it take a set of line segments $B=\{\overline{A_1A_2},\overline{A_2A_3},\overline{A_3A_4}\}$ and a set of lines $C=\{\overline{A_1A_2},\overline{A_2A_4}\}$

What special about these sets is that no two line segments (in each of these sets) have more then one point in common or you can say that no two line segments have another segment overlapping on each other (for example line segment $\overline{A_2A_4}$ and $\overline{A_1A_3}$ are overlapping on each other on line segment $\overline{A_2A_3}$), let these line segments be called dis-joining lines

So now lets define something :

If you are given $n$ points $A_1,A_2,A_3...,A_n$ all of which are co-linear and you have a set of line segments $B$, now if all pair of line segments that are in $B$ are dis-joining and all lines, if summed together, forms line segment $\overline{A_1A_n}$ (For example in set $C$ in above example $\overline{A_1A_2}+\overline{A_2A_4}=\overline{A_1A_4}$) then set $B$ is called a Good set, else it will be called a Bad set.

How many distinct Good sets sets are possible if $n$ co-linear points are given? (Give a general formula)

Note by Zakir Husain
4 months, 1 week ago

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One $n$ -sized line-1 set.
Two separate $x,x-n$-sized lines-$x=1/2/3/...n-1$, n-1 sets.
Three: $x,y,n-x-y$, where there are $1+2+3+...+n-2$ possibilities for $(x,y)$, $\dfrac{(n-2)(n-1)}{2}$ sets.
...
Sum them up for the answer.

- 4 months, 1 week ago

Umm...
$n+\sum _{i=1}^{n} \dfrac{\prod _{k=n} ^{i} (i-k)}{n}$

- 4 months ago

$n+ \sum_{i=1}^{n}\dfrac{∏_{k=n}^{i}(i-k)}{n}=n+ \sum_{i=1}^{n}\dfrac{(i-n)...\red{(i-i)}}{n}=n+ \sum_{i=1}^{n}\dfrac{0}{n}=n+0=n$

- 4 months ago

$n+\sum _{i=1}^{n} \dfrac{\prod _{k=1} ^{i} (n-k)}{i+1}$

- 4 months ago