3. Five couples want to have their pictures taken. In how many ways can they arrange themselves in a row if a. couples must stay together? b. they may stand anywhere? 4. There are 12 people in a dinner gathering. In how many ways can the host (one of the 12) arrange his guests around a dining table if a. they can sit on any of the chairs? b. 3 people insist on sitting beside each other? c. 2 people refuse to sit beside each other? 5. A teacher drew a number line on the board and named some points on it. The teacher then asked the class to list all the rays in the figure. A I student answered 30 rays in all and the teacher said it was correct. How many points were named in the figure?

To determine how many points were named on the number line, let's analyze the scenario step by step. ### Understanding Rays on a Number Line - **Definition of a Ray:** A ray on a number line is a half-line that starts at one point and extends infinitely in one direction. Each ray is uniquely determined by its starting point and its direction (either to the left or to the right). ### Calculating the Number of Rays 1. **Distinct Rays from Each Point:** - **Starting Point Selection:** For each point on the number line, you can create rays extending in both the left and right directions. - **Unique Directions:** Since the number line is one-dimensional, each point can have two distinct rays (left and right) except for the extreme points: - **Leftmost Point:** Can only extend a ray to the right. - **Rightmost Point:** Can only extend a ray to the left. 2. **Total Number of Rays:** - For **n** points on the number line: - **Middle Points:** Each of the \( n - 2 \) middle points can create 2 rays. - **Extreme Points:** The leftmost and rightmost points can each create 1 ray. - **Formula:** \( \text{Total Rays} = 2(n - 2) + 2 = 2n - 2 \) 3. **Given Information:** - The AI student correctly identified **30 rays**. - Setting up the equation: \( 2n - 2 = 30 \) 4. **Solving for n:** \[ 2n - 2 = 30 \\ 2n = 32 \\ n = 16 \] ### Conclusion **16 points** were named on the number line to produce a total of 30 distinct rays.