7 reindeer in a single file line, dasher cannot be next to prancer.

Question

overall, I can see that there would be 7! arrangements, but I don't know how to remove the instances where Dasher is next to Prancer. I can think of 12 arrangements of which dasher is next to prancer.

Answer 1

There are two possibilities of Dasher being on the end. Each of those has 5 possibilities for Prancer and then 5! options for the other 5 reindeer, for a total of 1200 possibilities with Dasher on the end. There are five possibilities of Dasher not on an end. Each has four possibilities for Prancer and then 5! options for the other reindeer, for a total of 2400 with Dasher somewhere in the middle. 3600 arrangements total

Answer 2

HINT: To obtain the number of arrangements where Dasher and Prancer are together, view them as a single unit. Think of all the ways that Dancer, Vixen, Comet, Cupid, Blitzen, and "DasherPrancer" can be arranged. Note that an arrangement where Dasher is on Prancer's left is distinct from an arrangement where Dasher is on Prancer's right, so you'll have to eliminate both sets of arrangements from your total count.