Question 1210161: King Arthur's round table has 8 evenly spaced chairs. In how many ways can 8 knights be seated in the chairs, if Sir Lancelot and Sir Gawain insist on being seated next to each other?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**1. Treat Lancelot and Gawain as a Single Unit**
Since Sir Lancelot and Sir Gawain insist on sitting next to each other, we can treat them as a single unit. This unit can be arranged in two ways: Lancelot-Gawain or Gawain-Lancelot.
**2. Arrange the Unit and the Remaining Knights**
Now, we have 7 entities to arrange around the circular table: the Lancelot-Gawain unit and the remaining 6 knights. The number of ways to arrange $n$ distinct objects in a circle is $(n-1)!$.
Thus, we have $(7-1)! = 6!$ ways to arrange these 7 entities.
**3. Account for the Arrangement of Lancelot and Gawain**
Since Lancelot and Gawain can switch places within their unit, we need to multiply the number of arrangements by 2.
**4. Calculate the Total Number of Arrangements**
Total number of arrangements = (arrangements of the unit and other knights) * (arrangements of Lancelot and Gawain)
Total number of arrangements = $6! \times 2$
$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
Total number of arrangements = $720 \times 2 = 1440$
**Therefore, there are 1440 ways for the 8 knights to be seated if Sir Lancelot and Sir Gawain insist on sitting next to each other.**
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