номер 110 гдз 8 класс алгебра Муравин Муравин

1)\begin{equation}A_{n}^{k}=A_{n-1}^{k}+k\cdot A_{n-1}^{k-1} \end{equation} \begin{equation}A_{n}^{k}=\frac{n!}{\left ( n-k \right )!} \end{equation} \begin{equation}A_{n-1}^{k}=\frac{\left ( n-1 \right )!}{\left ( n-1-k \right )!} \end{equation} \begin{equation}A_{n-1}^{k-1}=\frac{\left ( n-1 \right )!}{\left ( n-1-\left ( k-1 \right ) \right )}= \end{equation} \begin{equation}=\frac{\left ( n-1 \right )!}{\left ( n-1-k+1 \right )}=\frac{\left ( n-1 \right )!}{\left ( n-k \right )!} \end{equation} \begin{equation}\frac{n!}{\left ( n-k \right )!}=\frac{\left ( n-1 \right )!}{\left ( n-1-k \right )!}+k\frac{\left ( n-1 \right )!}{\left ( n-k \right )!} \end{equation} 2)\begin{equation}c_{n+1}^{k+1}=c_{n}^{k}+c_{n}^{k-1} \end{equation} \begin{equation}c_{n+1}^{k+1}=\frac{\left ( n+1 \right )!}{\left ( k+1 \right )!\left ( n+1-\left ( k+1 \right ) \right )!}= \end{equation} \begin{equation}=\frac{\left ( n+1 \right )!}{\left ( k+1 \right )!\left ( n-k \right )!} \end{equation} \begin{equation}c_{n}^{k}=\frac{n!}{k!\left ( n-k \right )!} \end{equation} \begin{equation}c_{n}^{k+1}=\frac{n!}{\left ( k+1 \right )!\left ( n-k-1 \right )!} \end{equation} \begin{equation}\frac{\left ( n+1 \right )!}{\left ( k+1 \right )!\left ( n-k \right )!}=\frac{n!}{k!\left ( n-k \right )!}+ \end{equation} \begin{equation}+\frac{n!}{\left ( k+1 \right )!\left ( n-k-1 \right )} \end{equation} 3)\begin{equation}C_{n}^{k}+2C_{n}^{k+1}+C_{n}^{k+2}=C_{n+2}^{k+2} \end{equation} \begin{equation}C_{n}^{k}=\frac{n!}{k!\left ( n-k \right )!} \end{equation} \begin{equation}C_{n}^{k+1}=\frac{n!}{\left ( k+1 \right )!\left ( n-k-1 \right )!} \end{equation} \begin{equation}C_{n}^{k+2}=\frac{n!}{\left ( k+2 \right )!\left ( n-k-2 \right )!} \end{equation} \begin{equation}C_{n+2}^{k+2}=\frac{\left ( n+2 \right )!}{\left ( k+2 \right )!\left ( n+2-k-2 \right )}= \end{equation} \begin{equation}=\frac{\left ( n+2 \right )!}{\left ( k+2 \right )!\left ( n-k \right )!} \end{equation} \begin{equation}\frac{n!}{k!\left ( n-k \right )!}+2\cdot \frac{n!}{\left ( k+1 \right )!\left ( n-k-1 \right )!}+ \end{equation} \begin{equation}+\frac{n!}{\left ( k+2 \right )!\left ( n-k-2 \right )!}=\frac{\left ( n+2 \right )!}{\left ( k+2 \right )!\left ( n-k \right )!} \end{equation}