Ten points are spaced equally around a circle. How many different chords can be formed by joining any 2 of these points? (A chord is a straight line joining two points on the circumference of a circle.)
(A) 9
(B) 45
(C) 17
(D) 66
(E) 55
Ten points are spaced equally around a circle. How many different chords can be formed by joining any 2 of these points? (A chord is a straight line joining two points on the circumference of a circle.)
(A) 9
(B) 45
(C) 17
(D) 66
(E) 55

## Solution

Space the ten points equally around the circle and label them $A_{1}, A_{2}, \ldots, A_{10}$ for convenience.

If we start with $A_{1}$ and join it to each of the other nine points, we will have 9 chords.

Similarly, we can join $A_{2}$ to each of the other 8 points. If we continue this process until we join $A_{9}$ to $A_{10}$ we will have $9+8+7+6+5+4+3+2+1=45$ chords.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-463.jpg?height=461&width=439&top_left_y=604&top_left_x=1323)

ANSWER: (B)