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Question Number 73982 by mhmd last updated on 17/Nov/19
$${if}\:{p}_{{r}} ^{{n}} =\mathrm{49}\:,\:{c}_{{r}} ^{{n}} =\mathrm{196}\:{find}\:{the}\:{valve}\:{of}\:{r}\:{and}\:{n}? \\ $$$${pleas}\:{sir}\:{help}\:{me} \\ $$
Commented by som(math1967) last updated on 17/Nov/19
$$\overset{{n}} {{c}}_{{r}} >\overset{{n}} {{p}}_{{r}} \:\:???{how}? \\ $$
Commented by Raxreedoroid last updated on 17/Nov/19
$${p}_{{r}} ^{{n}} ={r}!\centerdot{c}_{{r}} ^{{n}} \\ $$$$\mathrm{49}={r}!\centerdot\mathrm{196} \\ $$$${r}!=\frac{\mathrm{49}}{\mathrm{196}} \\ $$$${r}!=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\mathrm{solve}\:\mathrm{for}\:\mathrm{when}\:\Gamma\left({r}+\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{4}}? \\ $$$$\Gamma\left({x}\right)={c}\:\mathrm{range}\:\mathrm{when}\:\mathrm{x}\in\mathbb{R}\:\mathrm{is}\:\mathrm{almost}\:\mid{c}\mid>\mathrm{0}.\mathrm{88}...\left(\frac{\sqrt{\pi}}{\mathrm{2}}\right) \\ $$$$\mid\mathrm{0}.\mathrm{25}\mid<\mathrm{0}.\mathrm{8} \\ $$$$\mathrm{so}\:\mathrm{there}\:\mathrm{is}\:\mathrm{no}\:\mathrm{solution} \\ $$