Question Number 73983 by mhmd last updated on 17/Nov/19
$${if}\:{p}_{{r}} ^{{n}} =\mathrm{49}\:,\:{c}_{{r}} ^{{n}} =\mathrm{196}\:{find}\:{the}\:{valve}\:{of}\:{r}\:{and}\:{n}? \\ $$$${pleas}\:{sir}\:{help}\:{me} \\ $$
Answered by $@ty@m123 last updated on 17/Nov/19
$$\mathrm{4}×\:^{{n}} {P}_{{r}} \:=\:^{{n}} {C}_{{r}} \\ $$$$\Rightarrow\frac{\mathrm{4}.{n}!}{\left({n}−{r}\right)!}=\frac{{n}!}{{r}!\left({n}−{r}\right)!} \\ $$$$\Rightarrow\:{r}!=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$${Not}\:{possible}. \\ $$
Commented by JDamian last updated on 17/Nov/19
$$\:^{{n}} {P}_{{r}} \:<\:^{{n}} {C}_{{r}} ? \\ $$$${Why}\:\:\mathrm{4}×? \\ $$
Commented by $@ty@m123 last updated on 17/Nov/19
$$\because\:\mathrm{4}×\mathrm{49}=\mathrm{196} \\ $$
Answered by arkanmath7@gmail.com last updated on 17/Nov/19
$${r}!\:=\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$$${search}\:{in}\:{gamma}\:{function} \\ $$$${r}!\:=\:\Gamma\left({r}+\mathrm{1}\right)\:=\:{r}\Gamma\left({r}\right)\:=\:{r}\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} \:{x}^{{r}−\mathrm{1}} {dx} \\ $$