1. **Problem Statement:** Illustrate the sample space using a tree diagram for the orderings in which 3 girls Sarah, Tracy, and Beth may sit in a row of 3 chairs. 2. **Understanding the Problem:** We want to find all possible permutations of the 3 girls sitting in 3 chairs. Each chair can be occupied by one girl, and no girl can occupy more than one chair. 3. **Formula and Rules:** The number of ways to arrange $n$ distinct objects in order is given by the factorial: $$n! = n \times (n-1) \times (n-2) \times \cdots \times 1$$ For $n=3$, the number of permutations is: $$3! = 3 \times 2 \times 1 = 6$$ 4. **Constructing the Tree Diagram (conceptual):** - Step 1: Choose the girl for the first chair (3 options: Sarah, Tracy, Beth). - Step 2: For each choice in step 1, choose the girl for the second chair (2 remaining options). - Step 3: For each choice in step 2, choose the girl for the third chair (1 remaining option). 5. **Listing all permutations:** - Sarah, Tracy, Beth - Sarah, Beth, Tracy - Tracy, Sarah, Beth - Tracy, Beth, Sarah - Beth, Sarah, Tracy - Beth, Tracy, Sarah 6. **Explanation:** Each branch of the tree represents a choice for a chair. Starting from the first chair, we branch out to all possible girls, then for each branch, we branch out to the remaining girls for the second chair, and so on until all chairs are filled. **Final answer:** There are 6 possible orderings for Sarah, Tracy, and Beth to sit in a row of 3 chairs, as listed above.