Question

Four adults and four children are to be seated at a circular table. In how many different ways can they be seated if each child must be next to another child? (Two seatings are considered the same if one can be rotated to form the other.)

Answer 1

There are 144 different ways to seat four adults and four children at a circular table, with the constraint that each child must be next to another child.

Step-by-step explanation:

To determine how many different ways four adults and four children can be seated at a circular table with the condition that every child must be seated next to another child, consider the restriction as a single unit. This reduces the problem to arranging five units (four single adults and one unit of four children) in a circle. Since rotations of the same arrangement are considered identical in circular permutations, we effectively have one less unit to arrange. Thus, we arrange (5-1) units or 4 units in a straight line, which corresponds to (4-1)! permutations. Now, the four children can be arranged among themselves in 4! ways. Therefore, the total number of possible arrangements is 3! × 4!, which equals 6 × 24, or 144 different ways they can be seated.