## Geometry

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### Geometry

by swerve » Tue Jul 23, 2024 3:47 pm

00:00

A

B

C

D

E

## Global Stats

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

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### Re: Geometry

by malarkey » Wed Aug 07, 2024 8:24 am
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 $$\boxed{10}$$.

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