DSAverse

Initializing Sorting Algorithms...

Sorting

Trees

Graphs

Preparing interactive visualizations...

Back to Combinatorics

The Binomial Theorem

Multiplying out (a + b)ⁿ means picking a or b from every bracket. The coefficient of aⁿ⁻ᵏbᵏ counts the picks with k b's — and that is C(n, k).

Step 1 of 14Speed

(a + b)² is the area of a square with side a + b. Split each side into a piece of length a and a piece of length b, and the square falls into four rectangles.

(a + b)^{2} = \textcolor{#fbbf24}{a^{2}} + \textcolor{#4ade80}{2ab} + \textcolor{#38bdf8}{b^{2}} \\[4pt] (2 + 1)^{2} = 4 + 4 + 1 = \mathbf{9}

n (brackets)2Moving n switches back to the picks viewk (b's in the pick)0a2b1

Each term is coloured by its number of b's, as in the picture. The second line puts the numbers in: the terms always add up to (a + b)ⁿ.

The Binomial Theorem

Multiplying out (a + b)ⁿ means picking a or b from every bracket. The coefficient of aⁿ⁻ᵏbᵏ counts the picks with k b's — and that is C(n, k).

Step 1 of 14Speed

(a + b)² is the area of a square with side a + b. Split each side into a piece of length a and a piece of length b, and the square falls into four rectangles.

(a + b)^{2} = \textcolor{#fbbf24}{a^{2}} + \textcolor{#4ade80}{2ab} + \textcolor{#38bdf8}{b^{2}} \\[4pt] (2 + 1)^{2} = 4 + 4 + 1 = \mathbf{9}

n (brackets)2Moving n switches back to the picks viewk (b's in the pick)0a2b1

Each term is coloured by its number of b's, as in the picture. The second line puts the numbers in: the terms always add up to (a + b)ⁿ.

DSAverse

The Binomial Theorem

See also

The Binomial Theorem

See also