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
11 months, 1 week ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

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 - 11 months, 1 week ago

Log in to reply

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

Jeff Giff - 11 months ago

Log in to reply

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 - 11 months ago

Log in to reply

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

Jeff Giff - 11 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...