To determine the number of different angles that can be formed from 10 noncollinear rays with a common vertex, follow these steps:
1. Understanding Rays and Angles:
- Each pair of rays that emanate from the vertex can form an angle.
2. Calculation of Number of Pairs:
- To find how many unique pairs of rays can be formed from 10 rays, we need to calculate the combinations of 2 rays taken from 10. This is denoted as [tex]\(\binom{10}{2}\)[/tex], which is read as "10 choose 2".
3. Formula for Combinations:
- The formula for combinations is [tex]\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)[/tex], where [tex]\(n\)[/tex] is the total number of items to choose from, and [tex]\(k\)[/tex] is the number of items to choose.
- In this case, [tex]\(n = 10\)[/tex] and [tex]\(k = 2\)[/tex].
4. Application of Formula:
- [tex]\(\binom{10}{2} = \frac{10!}{2!(10-2)!}\)[/tex]
- Simplifying this, we get [tex]\(\binom{10}{2} = \frac{10 \times 9}{2 \times 1}\)[/tex]
- This further simplifies to 45.
5. Conclusion:
- Therefore, there are 45 different angles that can be formed by the 10 noncollinear rays emanating from the common vertex.
Final Answer: 45 different angles can be formed if there are 10 different noncollinear rays with a common vertex.
