person decided to play a game in which he wants all his 20 friends to sit along a circle. In how many ways all 20 friends is to be seated if 2
particular friends are always to sit together?
Start with simpler models. For example, 2 people in a row can be seated in 2 ways, but sitting along a circle, there is just 1 way. 3 people in a row can be seated in 6 ways, but once they are in a circle, there are just 2 ways (draw this to see why). Continue with 4 people and you will soon see the pattern you need for this more complex problem.
For the pair, consider them as one person, with 2 ways to seat them relative to each other.
can u solve it
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answers is 2/19
correct or not
To find the number of ways to seat all 20 friends in a circle where 2 particular friends always sit together, we can consider those 2 friends as a single entity.
Step 1: Treat the 2 particular friends as a single entity. So now we have 19 entities to seat in a circle.
Step 2: Arrange the remaining 19 entities in a circle. The number of ways to arrange n entities in a circle is (n-1)!. Therefore, in this case, there are (19-1)! = 18! ways to arrange the remaining 19 entities.
Step 3: Now, fix the position of the 2 particular friends within the arrangement obtained in step 2. Since the 2 friends are already considered as a single entity, there are 2 ways to arrange them within their fixed position.
Step 4: Multiply the number of ways from step 2 and step 3 to get the total number of seating arrangements.
Total number of seating arrangements = (number of ways from step 2) x (number of ways from step 3)
= 18! x 2
Therefore, there are 18! x 2 ways for all 20 friends to be seated in a circle if 2 particular friends always sit together.