OE

A circle has 10 points (A to J) on its circumference. We need to find the ratio of total
number of quadrilaterals to total number of hexagons that can be formed.


Quadrilaterals: Has four sides. Therefore, we need 4 points to form it. Total points are
10. Since order in which points are selected does not matter.

Since if any four points are selected only one quadrilateral can be formed
For example, if we consider 4 points say A, C, E and H then only one quadrilateral ACEH
can be formed.
Hence, the total number of quadrilaterals formed would be equal to the number of
groups of 4 points that can be selected.
Hence, we will apply combination.

The total number of combinations of 4 points or number of ways to select 4 points out
of 10 \(= C^{10}_4=\frac{10!}{4!6!}\)


Hexagons: Has six sides.
Similarly, if any six points are selected only one hexagon can be formed
For example, if we consider 6 points say A, B, C, D, E and H then only one quadrilateral
ABCDEH can be formed.
Hence the total number of hexagons formed would be equal to the number of groups of
6 points that can be selected.
Number of ways to choose that \(= C^{10}_6=\frac{10!}{6!4!}\)

The ratio is

\(\dfrac{\frac{10!}{4!6!}}{\frac{10!}{6!4!}}=1\)

C is the answer