The entries in the Pascal Triangle can, for n ≥ 2, be used to determine the number of k-sided figures that can be formed using a set of n points on a circle. In general, the first entry in a row indicates the number of n-sided figures that can be formed, the second entry indicates the number of (n − 1)-sided figures, and so on. For example, if a circle contains 4 points, the row for n = 4 in the Pascal Triangle shows the number of possible quadrilaterals (1), the number of triangles (4), and the number of line segments (6) that can be formed using
the four points.
(a) How many hexagons can be formed using 8 points lying on the circumference of a circle?
(b) How many triangles can be formed using 10 points lying on the circumference of a circle?
(c) How many dodecagons can be formed using 20 points lying on the circumference of a circle?