Binomial theorem

Japanese school year: Math II

What you learn

Learn the binomial theorem for expanding natural number powers of binomials and determining specific coefficients using combinations. This theorem is useful for extracting individual coefficients quickly and studying binomial distributions. Prior knowledge of algebraic expansion rules and combination calculations is recommended.

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Key points

This formula (the binomial theorem) expands $(a + b)^n$ all at once. The coefficient of each term can be calculated using the combination symbol ${}_n\mathrm{C}_r$ .

(a + b)^n = \sum_{r=0}^n {}_n\mathrm{C}_r a^{n-r} b^r

Use this formula for the general term to find a specific term without expanding the whole expression. Choose $r$ to match the power you need, and find its coefficient.

T_{r+1} = {}_n\mathrm{C}_r a^{n-r} b^r

This is a special form of the binomial theorem where $a$ is 1 and $b$ is $x$ . Since powers of 1 are always 1, each coefficient of $x^r$ is simply ${}_n\mathrm{C}_r$ .

(1 + x)^n = \sum_{r=0}^n {}_n\mathrm{C}_r x^r

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Practice 1Coming soon