Distinct points are placed on a circle. Each pair of points is joined with a line segment. An example with 4 points and 6 line segments is shown. If 8 distinct points are placed on a circle, how many line segments would there be?
(A) 27
(B) 36
(C) 21
(D) 56
(E) 28

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-056.jpg?height=252&width=247&top_left_y=2064&top_left_x=1443)
Solution 1

Label the 8 points $A$ through $H$ as shown and proceed to connect the points in all possible ways.

From point $A, 7$ line segments are drawn, 1 to each of the other points, $B$ through $H$.

From point $B, 6$ new line segments are drawn, 1 to each of the points $C$ through $H$, since the segment $A B$ has already been drawn.

This continues, with 5 line segments drawn from point $C, 4$ from $D$, 3 from $E, 2$ from $F, 1$ from $G$ and 0 from point $H$ since it will already

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-268.jpg?height=331&width=314&top_left_y=268&top_left_x=1640)
be joined to each of the other points.

In total, there are $7+6+5+4+3+2+1=28$ line segments.

## Solution 2

Label the 8 points $A$ through $H$ as shown above and proceed to connect the points in all possible ways.

From each of the 8 points, 7 line segments can be drawn leaving the point, 1 to each of the other 7 points.

Thus, the total number of line segments leaving the 8 points is $8 \times 7=56$.

However, this counts each of the line segments twice, since each segment will be counted as leaving both of its ends.

For example, the segment leaving point $A$ and ending at point $D$ is also counted as a segment leaving point $D$ and ending at point $A$.

Thus, the actual number of line segments is half of 56 or $56 \div 2=28$.

ANSWER: (E)