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Question Number 74000 by mhmd last updated on 17/Nov/19
$${if}\:{p}_{{r}} ^{{n}} =\mathrm{840}\:,\:{c}_{{r}} ^{{n}} =\mathrm{35}\:{find}\:{value}\:{of}\:{r}? \\ $$$${pleas}\:{sir}\:{help}\:{me}? \\ $$
Answered by MJS last updated on 17/Nov/19
$${C}_{{r}} ^{{n}} =\frac{{n}!}{{r}!\left({n}−{r}\right)!} \\ $$$${P}_{{r}} ^{\:{n}} =\frac{{n}!}{\left({n}−{r}\right)!}={C}_{{r}} ^{{n}} ×{r}! \\ $$$${C}_{{r}} ^{{n}} =\mathrm{35} \\ $$$${P}_{{r}} ^{\:{n}} =\mathrm{840} \\ $$$$\mathrm{35}{r}!=\mathrm{840} \\ $$$${r}!=\mathrm{24} \\ $$$${r}=\mathrm{4} \\ $$$$\Rightarrow \\ $$$$\frac{{n}!}{\left({n}−\mathrm{4}\right)!}=\mathrm{840} \\ $$$${n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\left({n}−\mathrm{3}\right)=\mathrm{840} \\ $$$${n}^{\mathrm{4}} −\mathrm{6}{n}^{\mathrm{3}} +\mathrm{11}{n}^{\mathrm{2}} −\mathrm{6}{n}−\mathrm{840}=\mathrm{0} \\ $$$$\left({n}−\mathrm{7}\right)\left({n}+\mathrm{4}\right)\left({n}^{\mathrm{2}} −\mathrm{3}{n}+\mathrm{30}\right)=\mathrm{0} \\ $$$${n}\geqslant\mathrm{0}\:\Rightarrow\:{n}=\mathrm{7} \\ $$