Number of straight lines that can be drawn from 20 points, of which 10 points are collinear.

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embizze eNotes educator| Certified Educator

We have 20 points, 10 of which are collinear. Assume that no other 3 points are collinear.

(1) For each of the 10 collinear points we can draw 11 lines -- the line they are contained within, and one line each through the remaining 10 points. Thus there are 10(11)=110 lines. But the line containing the points has been counted 10 times, so we must subtract the extra 9 to get 101 lines. (Or for each point there are 10 lines for 100 lines plus the collinear line itself to make 101.)

(2) For the 10 points not lying on the line: Each point makes a line with each of the other 19 points. But we have already counted the lines to the points on the 10pt collinear line. Thus we need only count lines between the 10 noncollinear points.

There are 10(9)=90 lines (each of the 10 points is connected to the 9 other points) but this counts each line twice (e.g. from A to B and from B to A) so we divide by 2 to get 45 additional lines.

The total number of lines is the sum 101+45=146 lines.