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 6 distinct points are placed on a circle, how many line segments would there be?
(A) 13
(B) 16
(C) 30
(D) 15
(E) 14
Solution 1

We label the 6 points $A$ through $F$ as shown and proceed to connect the points in all possible ways.

From point $A, 5$ line segments are drawn, 1 to each of the other points, $B$ through $F$.

From point $B, 4$ new line segments are drawn, 1 to each of the

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-261.jpg?height=274&width=314&top_left_y=199&top_left_x=1491)
points $C$ through $F$, since the segment $A B$ has already been drawn.

This continues, with 3 line segments drawn from point $C, 2$ from point $D, 1$ from point $E$, and 0 from point $F$ since it will have already been joined to each of the other points.

In total, there are $5+4+3+2+1=15$ line segments.

## Solution 2

Label the 6 points $A$ through $F$ as shown above.

From each of the 6 points, 5 line segments can be drawn leaving the point, 1 to each of the other 5 points.

Thus, the total number of line segments leaving the 6 points is $6 \times 5=30$.

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 $30 \div 2=15$.

ANSWER: (D)