You need to put your reindeer, jebediah, Lancer, Ezekiel, gloopin , In a single-file line to pull your sleigh. However , Lancer and Ezekiel are best friends, so you have to put them next to each other or they won't fly.

To ensure that Lancer and Ezekiel are next to each other in the single-file line to pull the sleigh, you can think about the line-up as a combination problem. Since they have to be next to each other, we can initially treat them as a single entity. Let's call this combined entity "L&E."

Now you have three separate entities: Jebediah, Gloopin, and "L&E" (which is Lancer and Ezekiel together). These three can be arranged in a line in 3! (3 factorial) ways, which is 3 x 2 x 1 = 6 different ways.

However, within the "L&E" entity, Lancer and Ezekiel can also switch places. Since there are two of them, they can be arranged in 2! (2 factorial) ways, which is 2 x 1 = 2 different ways.

To find the total number of ways you can line up Jebediah, Gloopin, Lancer, and Ezekiel in a single-file where Lancer and Ezekiel are next to each other, you multiply the arrangements of the three entities by the ways Lancer and Ezekiel can arrange themselves within their own entity. So the total number of arrangements is 6 (from Jebediah, Gloopin, and "L&E") times 2 (from Lancer and Ezekiel within "L&E"), which equals 12 different ways.

Extra: Understanding permutations and combinations is helpful for various real-world problems, such as this one. When dealing with permutations, the order matters, while with combinations, the order does not matter. In this case, we’re dealing with permutations because the order in which the reindeer line up affects the arrangement.

The term factorial (denoted by an exclamation point, i.e., n!) refers to the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

When faced with a constraint in a permutation problem—like having to keep Lancer and Ezekiel together—a good strategy is to group the constrained entities together and treat them as a single unit. After calculating the permutations for the larger grouping, you account for any permutations within the constrained group. This ensures you consider all possible configurations without breaking the constraint.

This kind of problem-solving is not limited to reindeer or holiday-themed scenarios. It can be applied to seat arrangements, lineup orders, or even more abstract problems like code ordering or organizing data in a particular order. Understanding how to deal with constraints can make these problems much easier to solve.