1. The problem statement, all variables and given/known data 5 dots in some plane can create how many lines? a)5 b)10 c)15 d)20 2. Relevant equations 3. The attempt at a solution If you ask me, I will say infinite, but the creators of the question think that one line can be created from at least 2 dots. Line can be created from 3, 4 ... n dots. 2 dots can create [tex]C_5^2[/tex] lines, and it is 10 lines. But what about those with 3 dots?
why infinite number of lines? I think you should just take 5 random points, and then count the number of lines.
Perhaps it would be better to say 5 points. A 'dot' is not a mathematical object. Of course, if the 5 points happened to lie on a single line, there would be only one line through them so the question really should be "What is the largest number of lines that can be drawn through 5 points in a plane?" Choosing any point, you can draw a line through it and any one of the 4 other points. That is, each point lies on 4 lines. If we multiply the number of points by the number of lines "per" point, we have 4(5)= 20. But each line accounts for 2 points so the correct answer is 20/2= 10. In fact, the largest number of lines that can be draw through n points in a plane is n(n-1)/2. Notice by the way that we can argue this in a slightly different way: choose one point (call it p1) out of the n points. You can draw lines through it and each of the other n-1 points: n-1 lines. Now pick another point (p2)( and draw lines through it and each of the other points, NOT counting p1. There are n- 2 "other points" (not p1 or p2) so n-2 such lines. Pick a third point, p3, and we can draw n- 3 lines through n-3 "other points" (not p1, p2, or p3). We can go through all of the points until we have used them all up and have (n-1)+ (n-2)+ (n-3)+ ...+ 3+ 2 +1 lines. That is the sum of the first n-1 positive integers which is well know to be n(n-1)/2. (It is perhaps better known that the sum of the first n integers is n(n+1)/2 but that immediately gives the first.)
HallsofIvy yes, I understand. But what I do not understand is that also line can be created by three points, or in that case the line will be repeated?
Yes, if three points happen to lie on a line the method I cite would count that same line three times. That was why I said the question should be "What is the largest number of lines that can be drawn through 5 points".
Yes. Ok, thank you. And what about this one: Five points in the space can create how many planes? a)10 b)8 c)5 e)4 One plane can be created with at least three points. So [tex]C_5^3[/tex]. The final answer is 10. But aren't there planes which are passing through 4 or 5 points? Or they will again be repeated?
You're asking for the maximum possible number of planes that can be formed by 5 points, not unlikely possibility that all 5 points may reside on the same plane. That's pretty much the same thing as the first question above.
The question reminds me of the Handshake problem. Read about THAT one. If you understand the solution methods, then this 5 Points problem is no more difficult than the Handshake problem.
Only if the "4 or 5 points" happen to lie on a single plane. In that case there would be fewer planes. That's a very special situation. You are clearly expected to find the maximum number of planes which would mean the points are in "general position". I.e. not 4 or more points lie on the same plane.