A mother, father, and their 3 children are having their picture taken. They will all be seated elbow-to-elbow on the living room couch, and the children will not be permitted to sit next to each other. How many different arrangements are possible for the picture?

Combinatorics with some restrictions

To solve this problem, we need to arrange the mother, father, and three children on a couch such that the children do not sit next to each other. Let’s denote the mother as M, the father as F, and the children as C1, C2, and C3.

Strategy

Position the Adults: Place M and F first. Since we need to ensure the children don’t sit next to each other, the adults must separate them. This creates gaps between and possibly around the adults to place the children. The arrangement of adults can be in 2 ways: MF or FM.

Distribute Children in Gaps: Considering adults are placed (e.g., MF), we can insert children into the gaps between and on the sides of the adults:

• One gap to the left of M
• One gap between M and F
• One gap to the right of F

So we have three gaps and three children to fill these gaps.

Permute the Children: Since no two children can be adjacent, each gap must be filled by exactly one child, and the children can…