+28 Four children and four adults are to be seated at a circular table. In how many different ways can they be seated if each child must be next to two adults? (Two seatings are considered the same if one can be rotated to form the other.)
+1234 Since children must be next to two adults, let's consider the children as a single unit. There are now 5 "units" to arrange around the table (4 adults + 1 unit of 4 children).
Here's how to solve the problem:
Total Arrangements without Restriction:
Initially, ignoring the child requirement, we can arrange 5 units (adults and children) around the circular table.
For circular arrangements of n distinct objects, there are (n-1)! ways.
In this case, there are (5-1)! = 4! = 24 ways to arrange the units.
Overcounting due to Rotation:
However, in a circular arrangement, rotating the entire configuration doesn't create a new seating order.
So, we've overcounted the arrangements by the number of ways to rotate 5 objects in a circle.
There are 5 positions, and rotating one slot to the right fills all the positions eventually.
Therefore, we've overcounted by a factor of 5.
Correcting for Overcounting:
To get the actual number of unique arrangements, we need to divide the initial arrangements by the overcounting factor:
Unique Arrangements = Total Arrangements / Overcounting Factor
Unique Arrangements = 24 arrangements / 5 rotations
Unique Arrangements = 4.8 (Since the answer deals with arrangements, we can't have a fraction of a seating)
Accounting for Child Arrangement:
So far, we've treated the children as a single unit. But within that unit, the 4 children can be arranged in 4! ways.
Final Answer:
To get the total number of arrangements where each child sits next to two adults, we multiply the number of unique arrangements for the units (adults and children) by the number of ways to arrange the children within their unit:
Total Arrangements = Unique Unit Arrangements * Child Arrangements
Total Arrangements = 4 * 4! = 4 * 24 = 96
Therefore, there are 96 ways to seat the children and adults around the table such that each child sits next to two adults.
