Question 476358

 "no three of which lie in the same straight line" truly mean that a line is defined by two points;all you need is two points to draw a line through them. Through three points you can draw three lines (imagine triangle, extend its sides to make lines)


here are steps how to get a formula {{{n(n-1)/2}}} which is your answer

1.
Draw, or suppose you have, n points in a plane. No three points lie in a straight line. You want to know how many lines can be drawn through two points at a time.

For example, you may have a circle with eight points, denoted A through H.

2.
Pick one point and determine how many pairs of points it can be in. If there are n points, the answer is n-1. This is how many lines can pass through that first point and another point at the same time.

Continuing with the above example, A can be matched up with B or C or D or E or F or G or H. That's seven possible matches.

3.

Pick the next point over. Its pairing with the first point has already been counted, but its pairing with the n-2 other points hasn't. Add n-2 to your earlier number, n-1, as possible lines through the points.

Continuing with the above example, B can have a line going through it and C through H. You don't count a line going through B and A, since you already did that in Step 2. So the possible lines through B are six.

4.

Continue with the pattern, adding n-3, then n-4, and so on. So the total sum of possible lines is n-1 + n-2 + n-3 + ... + 1. This is the same as summing up 1 + 2 + 3 + ... + n-1. It can be shown that the formula for 1 + 2 + 3 + ... + n-1 is:

 {{{n(n-1)/2}}}

Continuing with the above example, there were eight points, so n=8 gives a total number of possible lines through the points of n(n-1)/2 = 8*7/2 = 28. You can verify this yourself by adding the 7 found in Step 2 to the 6 found in Step 3 to 5, 4, 3, 2 and 1 to get 28. It also matches the result discussed in the introduction where the number of points was {{{n=3}}}: {{{n(n-1)/2 = 3*2/2 = 3}}} possible lines.

PS:

nadam se da je sada jasno Ivanka...:-)