w123lucy wrote:

Hi, I am having trouble understanding the solution to this question:

Denise wants to organize her media shelf with all her old DVDs. Her collection includes 8 from Harry Potter, 3 from Lord of the Rings, 3 from The Hobbit, and 4 from The Hunger Game series. She doesn't care about the order within each series, but they each need to be in their own group. Furthermore, she wants to make sure that The Lord of The Rings and The Hobbit series are beside each other. How many different ways can she arrange her DVDs?

Answer: 6! * 3! * 3! * 4! * 4! * 8!

My thought process: I understand that the 8! * 3! * 3! * 4! comes from rearranging the DVDs within each series. I guess I can see the 4! comes from rearranging the entire series with no constraints (correct me if I'm wrong on this). But I really don't understand where the 6! comes from. I would think that because LOTR and Hobbit have to be next to each other that we would be subtracting or dividing or some sort of cutting out a bunch of arrangements that don't have LOTR and Hobbit next to each other.

Reasoning the book gives for the 6!: To arrange the two groups (the entire series) that need to be beside each other, we have two ways to place these groups first and second, two ways to place these groups second and third, and two ways to place these groups third and fourth. This makes a total of 6 possibilities.

Is there another way someone can put this for me? I hope you understand my confusion.

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