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The binomial theorem

Expanding (a + b)ⁿ with Pascal's triangle and binomial coefficients.

(a + b)ⁿ expands to Σ C(n, k) aⁿ⁻ᵏ bᵏ: the coefficients are the row of Pascal's triangle, which are the same “n choose k” numbers that count selections. So a counting problem and an expansion problem are the same problem in two costumes.

Worked example: expand (a + b)^5

Algebra · Expanding

Expand (a + b)^5

\left(a + b\right)^{5}

Step by step

  1. \left(a + b\right)^{5}

    Start from the expression.

  2. a^{5} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + b^{5}

    Distribute (multiply out) the products.

Reveal the answer
a^{5} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + b^{5}

Try your own

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