Counting with repeats: arrangements, groups and stars and bars
Arrange marbles of the same colour in a row, count codes with repeated digits, split students into rooms and count flower bunches with stars and bars.
- Iran: grade 12
Guide
This studio follows Iranian grade 12 Discrete Mathematics, chapter 3 lesson 1 (arrangements with repeats, splitting into groups, selecting with repeats).
When some things are identical, swapping them does not make a new arrangement. If we pretend all are different, each real arrangement is counted several times, so we divide n! by the factorial of each group of identical things.
Common mistakes worked on here: counting identical things as different, not dividing by the factorials, forgetting rooms that are already filled, and counting the places of the bars only instead of the places of stars and bars together.