Combination of lines

There are 44 co-linear pints A1,A2,A3A_1,A_2,A_3 and A4A_4 Now suppose it take a set of line segments B={A1A2,A2A3,A3A4}B=\{\overline{A_1A_2},\overline{A_2A_3},\overline{A_3A_4}\} and a set of lines C={A1A2,A2A4}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 A2A4\overline{A_2A_4} and A1A3\overline{A_1A_3} are overlapping on each other on line segment A2A3\overline{A_2A_3}), let these line segments be called dis-joining lines

So now lets define something :

If you are given nn points A1,A2,A3...,AnA_1,A_2,A_3...,A_n all of which are co-linear and you have a set of line segments BB, now if all pair of line segments that are in BB are dis-joining and all lines, if summed together, forms line segment A1An\overline{A_1A_n} (For example in set CC in above example A1A2+A2A4=A1A4\overline{A_1A_2}+\overline{A_2A_4}=\overline{A_1A_4}) then set BB is called a Good set, else it will be called a Bad set.

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

Note by Zakir Husain
3 weeks, 3 days ago

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One nn -sized line-1 set.
Two separate x,xnx,x-n-sized lines-x=1/2/3/...n1x=1/2/3/...n-1, n-1 sets.
Three: x,y,nxyx,y,n-x-y, where there are 1+2+3+...+n21+2+3+...+n-2 possibilities for (x,y)(x,y), (n2)(n1)2\dfrac{(n-2)(n-1)}{2} sets.
...
Sum them up for the answer.

Jeff Giff - 3 weeks, 3 days ago

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

Jeff Giff - 2 weeks, 3 days ago

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n+i=1nk=ni(ik)n=n+i=1n(in)...(ii)n=n+i=1n0n=n+0=nn+ \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

Zakir Husain - 2 weeks, 3 days ago

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

Jeff Giff - 2 weeks, 3 days ago

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