Problem Solving

How many different line segments can be drawn to connect all possible pairs of points shown above?

A. \(32\)
B. \(25\)
C. \(20\)
D. \(10\)
E. \(5\)

The OA is D

To determine how many different line segments can be drawn to connect all possible pairs of 5 points, you need to calculate the number of ways to choose 2 points out of 5 to form a line segment. This is a combinatorial problem that can be solved using combinations.

The number of ways to choose 2 points out of 5 is given by the combination formula:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

where \( n \) is the total number of points and \( k \) is the number of points to choose.

For this problem:

- \( n = 5 \) (the total number of points)
- \( k = 2 \) (since a line segment is defined by 2 points)

Applying the formula:

\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!}
\]

Simplify the factorials:

\[
\frac{5!}{2! \cdot 3!} = \frac{5 \times 4 \times 3!}{2! \times 3!} = \frac{5 \times 4}{2!} = \frac{5 \times 4}{2 \times 1} = 10
\]

Thus, the number of different line segments that can be drawn to connect all possible pairs of 5 points is \( \boxed{10} \).