Binomio de Newton (Potencia n-ésima de un binomio)
$$
(a + b)^n = \binom{n}{0} a^{n}b^{0} + \binom{n}{1} a^{n-1}b^{1} + \binom{n}{2} a^{n-2}b^{2} + \cdots + \binom{n}{n-2} a^{2}b^{n-2} + \binom{n}{n-1} a^{1}b^{n-1} + \binom{n}{n} a^{0}b^{n}
$$
$$
(a - b)^n = \binom{n}{0} a^{n}b^{0} - \binom{n}{1} a^{n-1}b^{1} + \cdots + (-1)^{n-1}\binom{n}{n-1} a^{1}b^{n-1} + (-1)^n\binom{n}{n} a^{0}b^{n}
$$
Números Combinatorios
$$\left\{\begin{array}{l}
\mathbf{\hbox{(Factorial de n) }} n! = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1 \hbox{ , } \forall n \in \mathbb{N}\\
\\
\mathbf{\hbox{(Propiedades)}} \left\{\begin{array}{l}
n! = n \cdot (n-1)! \\
\\
0! = 1 \\
\end{array}\right.\\
\\
\mathbf{\hbox{(Número combinatorio) }} \hbox{Dados n, m} \in \mathbb{N},n \geq m, \binom{n}{m} = \frac{n!}{m!(n-m)!} \\
\\
\mathbf{\hbox{(Propiedades)}} \left\{\begin{array}{l}
\binom{n}{0} = 1, \binom{n}{1} = n, \binom{n}{n} = 1 \\
\\
\binom{n}{m} = \binom{n}{n-m} \\
\\
\binom{n}{m} + \binom{n}{m+1} = \binom{n+1}{m+1} \\
\end{array}\right.\\
\end{array}\right.
$$
Triángulo de Tartaglia (o de Pascal)
$$
\left.\begin{array}{l}
\left.\begin{array}{l}
& & & & 1 & & & & \\
& & & 1 & & 1 & & & \\
& & 1 & & 2 & & 1 & & \\
& 1 & & 3 & & 3 & & 1 & \\
1 & & 4 & & 6 & & 4 & & 1\\
& & & & \vdots & & & & \\
\end{array}\right.
& \Leftrightarrow &
\left.\begin{array}{l}
& & & & \binom{0}{0} & & & & \\
& & & \binom{1}{0} & & \binom{1}{1} & & & \\
& & \binom{2}{0} & & \binom{2}{1} & & \binom{2}{2} & & \\
& \binom{3}{0} & & \binom{3}{1} & & \binom{3}{2} & & \binom{3}{3} & \\
\binom{4}{0} & & \binom{4}{1} & & \binom{4}{2} & & \binom{4}{3} & & \binom{4}{4}\\
& & & & \vdots & & & & \\
\end{array}\right.\\
\end{array}\right.
$$
Otras identidades (y sus aplicaciones)
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$$\left.\begin{array}{l}
\left\{\begin{array}{l}
(a - b)=(\sqrt[3]{a} - \sqrt[3]{b}) \cdot (\sqrt[3]{a^2} + \sqrt[3]{ab} + \sqrt[3]{b^2}) \\
(a + b)=(\sqrt[3]{a} + \sqrt[3]{b}) \cdot (\sqrt[3]{a^2} - \sqrt[3]{ab} + \sqrt[3]{b^2}) \\
\end{array}\right. & \mathbf{\hbox{(Indeterminaciones en límites con raíces cúbicas)}} \\
\\
\left\{\begin{array}{l}
(a - b)=(\sqrt[4]{a} - \sqrt[4]{b}) \cdot (\sqrt[4]{a^3} + \sqrt[4]{a^2b} + \sqrt[4]{ab^2} + \sqrt[4]{b^3}) \\
(a + b)=(\sqrt[4]{a} + \sqrt[4]{b}) \cdot (\sqrt[4]{a^3} - \sqrt[4]{a^2b} + \sqrt[4]{ab^2} - \sqrt[4]{b^3}) \\
\end{array}\right. & \mathbf{\hbox{(Indeterminaciones en límites con raíces cuartas)}} \\
\end{array}\right.$$
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Binomio de Newton (Clica para mostrar)
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