二項定理の公式
（一般化）二項定理
$$ (a+b)^n=\sum_{r=0}^{n} {}_n \mathrm{C}_r a^{n-r}b^{n} $$
例: 2乗の場合
$$ \begin{aligned} (a+b)^2 &= \sum_{r=0}^{2} {}_2 \mathrm {C}_r a^{2-r}b^r \\ &= {}_2 \mathrm{C}_0 a^2b^0 + {}_2 \mathrm{C}_1 a^1b^1 + {}_2 \mathrm{C}_2 a^0b^2 \\ &= a^2 + 2ab + b^2 \end{aligned} $$
例: 3乗の場合
$$ \begin{aligned} (a+b)^3 &= \sum_{r=0}^{3} {}_3 \mathrm{C}_r a^{3-r}b^r \\ &= {}_3 \mathrm{C}_0 a^3b^0 + {}_3 \mathrm{C}_1 a^2b^1 + {}_3 \mathrm{C}_2 a^1b^2 + {}_3 \mathrm{C}_3 a^0b^3 \\ &= a^3 + 3a^2b + 3ab^2 + b^3 \end{aligned} $$