An elf, a dwarf, a hobbit, a wizard, and a man are sitting in a line. How many different ways can they sit if the elf and the dwarf insist on sitting next to each other?

Consider the elf and the dwarf as one item, so you are now arranging 4

which is 4! or 24
but we can switch the elf and the dwarf and still have them side by side, so

2(4!) = 48

To solve this problem, we can treat the elf and the dwarf as a single entity. So we will have four entities in total: the combined elf-dwarf, hobbit, wizard, and man. We can arrange these four entities in 4! (4 factorial) ways.

However, the elf and the dwarf can also be arranged among themselves in 2! ways (since there are only two of them). Therefore, we need to multiply the 4! by 2! to account for the different ways they can sit within the combined entity.

So the total number of different ways they can sit with the elf and the dwarf sitting next to each other is:

4! × 2! = 24 × 2 = 48 ways.

To find the number of different ways the elf and the dwarf can sit next to each other, we can treat them as a single entity. This means we have four individuals: the elf-dwarf pair, the hobbit, the wizard, and the man.

Now, place the elf-dwarf pair together, so you have E-D, H, W, M. Since there are four entities, we can arrange them in 4! (4 factorial) ways.

However, within the elf-dwarf pair, the elf and the dwarf can switch places, so we need to multiply the result by 2.

Therefore, the total number of different ways the elf and the dwarf can sit next to each other is 4! * 2 = 24 * 2 = 48.

mmkmk