The block method (“must sit together”)
When a group of items must stay together, glue them into a single “super-item.” Arrange the super-item along with everything else as usual, then multiply by the number of ways to arrange the items within the block. For example, arranging people in a row where 3 specific people must sit together: treat the block of 3 as one unit, giving arrangements of units, times internal arrangements of the block.
The gaps method (“must not be adjacent”)
When certain items must not be adjacent to each other, arrange everyone else first, which creates a fixed number of gaps (including the two ends) between and around them. Then place the restricted items into those gaps, at most one per gap, so they're automatically separated.
Circular seating and reflection symmetry
For circular arrangements, use instead of (rotations of the same arrangement aren't counted as different). If the arrangement is also considered the same under reflection (like a bracelet, viewable from either side), divide by an additional factor of 2:.
Worked example
Problem: 6 people, including a couple who refuse to sit next to each other, are seated in a row. How many valid arrangements are there?
Solution: Total arrangements of 6 people: . Arrangements where the couple does sit together: treat them as one block, giving arrangements of 5 units, times for the couple's internal order: .
Subtract (complement counting): valid arrangements.
Practice problems
1.In how many ways can 5 people be seated in a row if 2 specific people must sit together?
2.In how many ways can 4 boys and 3 girls be seated in a row so that no two girls are adjacent?
3.In how many distinct ways can 8 people be seated around a circular table?
4.How many distinct bracelets can be made from 6 different beads, if the bracelet can be flipped over?