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# Combinatorics

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Elrond is organizing a council in Rivendell. He has invited two men, two dwarves, and two elves, and plans to seat them along with himself in a circle with seven seats. He knows that dwarves and elves do not get along, so he plans to seat them so that no dwarf is sitting next to an elf. The circle has a head chair. If Elrond (himself an elf) must sit at the head of the circle, in how many ways can he seat the other six guests? All beings are distinguishable from one another, and the head chair is distinguishable from the other chairs.

Aug 18, 2019

$$\text{If you study this you find that you must have the pattern \\MDDM embedded in the seating arrangement}\\ \text{There are 3 slots in which these 4 guests can be fit}\\ \text{Having fit them the 2 elves sit in the remaining seats}\\ \text{There are 4 ways to arrange MDDM}\\ \text{There are 2 ways to arrange the elves once the others are seated}\\ N = 3 \cdot 4 \cdot 2 = 24$$