1. **State the problem:** We have 24 points placed around a circle, and we want to find how many line segments are drawn between every pair of points. 2. **Formula used:** The number of line segments formed by connecting every pair of $n$ points is given by the combination formula: $$\text{Number of line segments} = \binom{n}{2} = \frac{n(n-1)}{2}$$ 3. **Explanation:** This formula counts the number of ways to choose 2 points out of $n$ to form a line segment. Since each pair of points forms exactly one line segment, this gives the total number of line segments. 4. **Apply the formula:** For $n=24$ points, $$\binom{24}{2} = \frac{24 \times (24-1)}{2} = \frac{24 \times 23}{2}$$ 5. **Calculate:** $$\frac{24 \times 23}{2} = 12 \times 23 = 276$$ 6. **Final answer:** There are **276** line segments drawn between the 24 points around the circle.