1. **Problem statement:** We need to find the number of ways to arrange 4 reindeer — Lancer, Prancer, Gloopin, and Bloopin — in a single-file line such that Lancer and Prancer are not next to each other. 2. **Total arrangements without restriction:** There are 4 reindeer, so the total number of ways to arrange them is $$4! = 24$$. 3. **Calculate arrangements where Lancer and Prancer are together:** - Treat Lancer and Prancer as a single unit. - Then we have this unit plus Gloopin and Bloopin, so 3 units total. - Number of ways to arrange these 3 units is $$3! = 6$$. - Inside the unit, Lancer and Prancer can be arranged in $$2! = 2$$ ways. - So total arrangements with Lancer and Prancer together is $$3! \times 2! = 6 \times 2 = 12$$. 4. **Calculate arrangements where Lancer and Prancer are apart:** - Subtract the number of arrangements where they are together from the total arrangements: $$24 - 12 = 12$$. 5. **Final answer:** There are $$12$$ ways to arrange the reindeer so that Lancer and Prancer are not next to each other.