Binomial theorem

Expand (a + b)ⁿ without multiplying — Pascal's triangle does the work.

Expand (a + b)ⁿ without multiplying brackets out — Pascal's triangle gives the coefficients.

Try this

power n

Pascal's row n = coefficients of (a + b)ⁿ = 1 4 6 4 1

Binomial coefficient

C (n, k) = n! / (k! \cdot (n - k)!)

This is the entry in row n, position k of Pascal's triangle — and the number of ways to choose k items from n.

Your turn

Expand (x + 2)³.

Watch out

Forgetting to raise the *whole* second term, including its sign. In (x − 3)², the last term is (−3)² = +9, not −9.

Theorem

(a + b)^{n} = \sum C (n, k) a^{n - k} b^{k}

Try it

(a+b)³

a³ + 3a²b + 3ab² + b³ — coefficients from Pascal's row 1, 3, 3, 1.