Question 1210162: Now suppose that not only must Sir Lancelot and Sir Gawain be next to each other, but Sir Galahad and Sir Percival also demand to be next to each other. How many seatings of the 8 knights are possible?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step by step:
**1. Treat Lancelot and Gawain as a Unit, and Galahad and Percival as a Unit**
* Since Lancelot and Gawain must sit together, we treat them as a single unit (LG or GL).
* Similarly, since Galahad and Percival must sit together, we treat them as another single unit (GP or PG).
**2. Arrange the Units and Remaining Knights**
* Now we have 4 units to arrange: the Lancelot-Gawain unit, the Galahad-Percival unit, and the 4 remaining knights.
* The number of ways to arrange $n$ distinct objects in a circle is $(n-1)!$.
* So, we have $(6-1)! = 5!$ ways to arrange these 6 units around the round table.
**3. Account for the Arrangements Within the Units**
* Lancelot and Gawain can switch places within their unit, so there are 2 ways to arrange them (LG or GL).
* Galahad and Percival can also switch places within their unit, so there are 2 ways to arrange them (GP or PG).
**4. Calculate the Total Number of Arrangements**
* To get the total number of arrangements, we multiply the number of ways to arrange the units by the number of ways to arrange the knights within each unit:
* Total arrangements = $5! \times 2 \times 2$
**5. Perform the Calculations**
* $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
* Total arrangements = $120 \times 2 \times 2 = 480$
**Therefore, there are 480 possible seatings of the 8 knights if Sir Lancelot and Sir Gawain are next to each other, and Sir Galahad and Sir Percival are also next to each other.**
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