Three dwarves and three elves sit down in a row of six chairs. If no dwarf will sit next to another dwarf and no elf wil sit next to another elf, in how many different ways can the elves and dwarves sit?

According to the book, the only way to ensure that no two dwarves and no two elves sit next to each other is to have the dwarves and elves alternate seats (DEDEDE or EDEDED). .....(I understood the rest of the explanation).
The anwer is 6*2*1*3*2*1=72.

Why is the anwer not 72*2=144? I think 72 should be multiplied by 2 because there are two ways of seating (DEDEDE or EDEDED). Thank you.

The 6 takes care of that. 6 means you have 6 ways of choosing from D AND E (not just D or E)

Yep - the first number in the multiplication sequence allows for all 6 possibilities to start, which means we are counting ways to start with D and we are counting ways to start with E.