How many lines are determined by 8 points, no three of which are collinear? How many triangles are determined by the same points if no four are coplanar? Solution The first part of the question uses combinations. Since running a line through points A and B makes no difference from running a line through points B and A, we know this is a combination problem (not a permutation problem) since order does not matter. Therefore, 2 Choose 8 is the number of lines determined by 8 points, since a line consists of only 2 points. The second part of the question uses the same concept, only now, 3 points are involved (consisting of the triangle\'s vertices.) 3 Choose 8. .

1. How many lines are determined by 8 points, no three of which are collinear?
How many triangles are determined by the same points if no four are coplanar?
Solution
The first part of the question uses combinations. Since running a line through points A and B
makes no difference from running a line through points B and A, we know this is a combination
problem (not a permutation problem) since order does not matter. Therefore, 2 Choose 8 is the
number of lines determined by 8 points, since a line consists of only 2 points. The second part of
the question uses the same concept, only now, 3 points are involved (consisting of the triangle's
vertices.) 3 Choose 8.