The binomial theorem and Pascal's triangle
See (a + b)³ as three brackets and count the paths that give each term, read the coefficients from Pascal's triangle, and use them for expansions with numbers.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- UK: Year 12
- France: grade 12
- Afghanistan: grade 11
- Syria: grade 11
- Palestine: grade 10
- Russia: grade 10
Guide
This studio follows Afghan grade 11 chapter 9 (the binomial theorem and Pascal's triangle) and the UK A-level binomial expansion.
In (a + b)3 each of the three brackets gives an a or a b. A path in the tree is one such choice. The coefficient of a2b is the number of paths with two a's and one b: the number of ways to choose which bracket gives the b.
Common mistakes worked on here: counting letters instead of paths, forgetting the numbers inside the brackets, taking a neighbouring term of Pascal's row, and thinking the coefficients add up to n.