# 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
3 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.

- 3 years, 7 months ago

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

- 3 years, 7 months ago

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

- 3 years, 7 months ago

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.

- 3 years, 6 months ago

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)}

- 3 years, 6 months ago

Solution needed.

- 3 years, 7 months ago