Binomial coefficients, Pascal's triangle and proof by counting
Split groups of friends into two kinds, prove Pascal's rule by counting, and see why each row adds up to a power of 2.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- France: grade 12
- Iran: grade 12
- Russia: grade 9
Guide
This studio follows Iranian grade 12 Discrete Mathematics, chapter 3 lesson 1 (combinations revisited) and the Khayyam-Pascal triangle.
In a combinatorial proof we count one number in two ways. If both ways are right, the results are equal. For example we count the groups of three among 6 friends once altogether and once in two kinds: the groups with Sara and the groups without her.
Common mistakes worked on here: counting order for a group, not adding the two kinds, multiplying instead of adding, and forgetting the empty group.