Question Number 182437 by akolade last updated on 09/Dec/22
Answered by qaz last updated on 09/Dec/22
$$\begin{pmatrix}{{n}+\mathrm{2}}\\{\:\:\:\:{r}}\end{pmatrix}=\left[{x}^{{r}} \right]\left(\mathrm{1}+{x}\right)^{{n}+\mathrm{2}} =\left[{x}^{{r}} \right]\left(\left(\mathrm{1}+{x}\right)^{{n}} +\mathrm{2}{x}\left(\mathrm{1}+{x}\right)^{{n}} +{x}^{\mathrm{2}} \left(\mathrm{1}+{x}\right)^{{n}} \right) \\ $$$$=\left[{x}^{{r}} \right]\left(\mathrm{1}+{x}\right)^{{n}} +\mathrm{2}\left[{x}^{{r}−\mathrm{1}} \right]\left(\mathrm{1}+{x}\right)^{{n}} +\left[{x}^{{r}−\mathrm{2}} \right]\left(\mathrm{1}+{x}\right)^{{n}} \\ $$$$=\begin{pmatrix}{{n}}\\{{r}}\end{pmatrix}+\mathrm{2}\begin{pmatrix}{\:\:\:{n}}\\{{r}−\mathrm{1}}\end{pmatrix}+\begin{pmatrix}{\:\:\:{n}}\\{{r}−\mathrm{2}}\end{pmatrix}\:\:\:\:\:,{by}\:{egorychev}'{s}\:{method} \\ $$
Commented by akolade last updated on 17/Dec/22
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much} \\ $$