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5 super hard questions....

  1. Suppose we have n distinct lines on a plane. What is the maximum and minimum number of intersections?

  2. Suppose we have n distinct points on a plane. What is the maximum and minimum number of line that they can form?

  3. What is the maximum number of region that n lines can divide a plane into?

  4. Given n collinear points , what is the maximum and minimum distance between the points?

  5. Arrange seven points on a plane such that , for any 3 points , we can find 2 points with distance 1.

Note by Werjohi Khhun
2 years, 7 months ago

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  1. Minimum=0, all lines are parallel.For maximum, no line must be parallel or concurrent. Now each of n lines intersects at 1 point. Thus, maximum number of intersections=\(\dbinom{n}{2}\)

  2. Minimum=1, all points are collinear. For maximum, assume all points on a circle, ie. no 3 of them are collinear, this will give again \(\dbinom{n}{2}\)

  3. This can be proved by recursion. See complete proof here

  4. Minimum\(=0^+\), maximum\(=\infty\). Either I am not understanding it properly or its vague.

Pranjal Jain - 2 years, 7 months ago

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For 1 , 2 , how do u know that it is N 2 ?? And , can you please explain what's that……??

Werjohi Khhun - 2 years, 7 months ago

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How many ways are there to select \(2\) objects from \(N\) given objects? That's \(\dbinom{N}{2}\). Thus, in \(1.\), I selected \(2\) lines out of \(n\) to find number of intersections. Similarly, in \(2.\), I selected \(2\) points out of \(n\) points, to draw a line.

  • In general, \(\dbinom{x}{y}\) is the number of ways to select \(y\) objects out of \(x\) given objects. For more details, see this. I hope it helps.

  • Though if you don't know about Permutations as well, I'll recommend you to read a wiki on permuations from here first.

  • \(\dbinom{N}{2}=\dfrac{N(N-1)}{2}\)

  • If you need some more help, feel free to either comment here or at Moderator's Messageboard here

Pranjal Jain - 2 years, 7 months ago

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@Pranjal Jain Oh , i never learnt about that tyep of permutations.... But why isit 2 ?? In what situation we can use 3,4 or 5 ?

Thank you.

Werjohi Khhun - 2 years, 6 months ago

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@Werjohi Khhun When you are supposed to select \(3\) objects out of \(N\) objects, use \(3\).

Say, we are given \(5\) objects, \((A,B,C,D,E)\)

  • Number of ways to choose \(2\) objects:\(\dbinom{5}{2}=10\)

{(A,B),(A,C),(A,D),(A,E),(B,C),(B,D),(B,E),(C,D),(C,E),(D,E)}

  • Number of ways to choose \(4\) objects:\(\dbinom{5}{4}=5\)

{(A,B,C,D),(A,B,C,E),(A,B,D,E),(A,C,D,E),(B,C,D,E)}

  • Number of ways to select \(5\) objects:\(\dbinom{5}{5}=1\)

{(A,B,C,D,E)}

Pranjal Jain - 2 years, 6 months ago

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Solution needed.

Werjohi Khhun - 2 years, 7 months ago

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