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 \( \boxed{10} \).
