Potencias de un binomio
$$
\mathbf{\hbox{(Casos particulares)}}
\left\{\begin{array}{l}
(a + b)^1 = (a + b) = a + b \\
(a - b)^1 = (a - b) = a - b \\
(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2 \\
(a - b)^2 = (a - b)(a - b) = a^2 - 2ab + b^2 \\
(a + b)^3 = (a + b)(a + b)(a + b) = (a + b)^2(a + b) = a^3 + 3a^2b + 3ab^2 + b^3 \\
(a - b)^3 = (a - b)(a - b)(a - b) = (a - b)^2(a - b) = a^3 - 3a^2b + 3ab^2 - b^3 \\
\end{array}\right.\\
$$
Binomio de Newton
$$
(a + b)^n = \binom{n}{0} a^{n}b^{0} + \binom{n}{1} a^{n-1}b^{1} + \binom{n}{2} a^{n-2}b^{2} + \cdots + \binom{n}{n-2} a^{2}b^{n-2} + \binom{n}{n-1} a^{1}b^{n-1} + \binom{n}{n} a^{0}b^{n}
$$
$$
(a - b)^n = \binom{n}{0} a^{n}b^{0} - \binom{n}{1} a^{n-1}b^{1} + \cdots + (-1)^{n-1}\binom{n}{n-1} a^{1}b^{n-1} + (-1)^n\binom{n}{n} a^{0}b^{n}
$$
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)
$$
\left.\begin{array}{l}
\left.\begin{array}{l}
& & & & 1 & & & & \\
& & & 1 & & 1 & & & \\
& & 1 & & 2 & & 1 & & \\
& 1 & & 3 & & 3 & & 1 & \\
1 & & 4 & & 6 & & 4 & & 1\\
& & & & \vdots & & & & \\
\end{array}\right.
& \Leftrightarrow &
\left.\begin{array}{l}
& & & & \binom{0}{0} & & & & \\
& & & \binom{1}{0} & & \binom{1}{1} & & & \\
& & \binom{2}{0} & & \binom{2}{1} & & \binom{2}{2} & & \\
& \binom{3}{0} & & \binom{3}{1} & & \binom{3}{2} & & \binom{3}{3} & \\
\binom{4}{0} & & \binom{4}{1} & & \binom{4}{2} & & \binom{4}{3} & & \binom{4}{4}\\
& & & & \vdots & & & & \\
\end{array}\right.\\
\end{array}\right.
$$
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