Subjects combinatorics

Reindeer Arrangement 21D536

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1. **Problem statement:** We need to arrange 5 reindeer: Gloopin, Quentin, Ezekiel, Lancer, and one more unnamed reindeer in a single line. 2. **Condition:** Quentin and Gloopin must be next to each other. 3. **Approach:** Treat Quentin and Gloopin as a single combined unit since they must be together. 4. **Step 1:** Combine Quentin and Gloopin into one block. Now we have this block plus Ezekiel, Lancer, and the other reindeer, making 4 units total. 5. **Step 2:** The number of ways to arrange these 4 units is $4! = 24$. 6. **Step 3:** Inside the combined block, Quentin and Gloopin can be arranged in $2! = 2$ ways (Quentin first or Gloopin first). 7. **Step 4:** Multiply the arrangements: total ways = $4! \times 2! = 24 \times 2 = 48$. **Final answer:** There are **48** ways to arrange the reindeer so that Quentin and Gloopin are next to each other.