Question Number 161888 by henderson last updated on 23/Dec/21
$$\mathrm{help}\:\mathrm{me}\:! \\ $$$$\mathrm{solve}\:\mathrm{this}\:\mathrm{one}\::\:\mathrm{C}_{\mathrm{40}} ^{\mathrm{2n}} \:=\:\mathrm{C}_{\mathrm{40}} ^{\mathrm{16}+\mathrm{n}} \\ $$
Commented by mr W last updated on 23/Dec/21
$${C}_{{r}} ^{{k}} ={C}_{{r}} ^{{r}−{k}} \\ $$$$\mathrm{2}{n}=\mathrm{40}−\left(\mathrm{16}+{n}\right) \\ $$$$\Rightarrow{n}=\mathrm{8} \\ $$
Commented by greg_ed last updated on 23/Dec/21
$$\mathrm{thank}\:\mathrm{you},\:\mathrm{sir}\:\mathrm{W}\:! \\ $$
Commented by kapoorshah last updated on 24/Dec/21
$${wrong}!!! \\ $$$${k}\:\geqslant\:{r} \\ $$$$ \\ $$
Commented by mr W last updated on 24/Dec/21
$${don}'{t}\:{say}\:{wrong}!\:{it}'{s}\:{a}\:{question}\:{of} \\ $$$${definition}.\:{not}\:{all}\:{countries}\:{use} \\ $$$${the}\:{same}\:{definition}.\:{in}\:{some}\:{countries} \\ $$$${like}\:{france},\:{russia},\:{polen},\:{china}\:{etc}. \\ $$$${people}\:{write}\:{C}_{{n}} ^{{k}} \:{instead}\:{of}\:{C}_{{k}} ^{{n}} \:{for} \\ $$$${the}\:{same}\:{thing}\:\frac{{n}!}{{k}!\left({n}−{k}\right)!}. \\ $$
Commented by Ar Brandon last updated on 24/Dec/21
$$\mathrm{Mr}\:\mathrm{W}\:\mathrm{is}\:\mathrm{right}\: \\ $$$$\mathrm{C}_{{n}} ^{{k}} =\overset{{n}} {\:}\mathrm{C}_{{k}} =\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}=\frac{{n}!}{{k}!\left({n}−{k}\right)!} \\ $$
Answered by Ar Brandon last updated on 23/Dec/21
$$\mathrm{2}{n}=\mathrm{16}+{n}\Rightarrow{n}=\mathrm{16} \\ $$$${C}_{\mathrm{40}} ^{\mathrm{2}\left(\mathrm{16}\right)} =\mathrm{C}_{\mathrm{40}} ^{\mathrm{16}+\mathrm{16}} \\ $$