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Question Number 102269 by mohammad17 last updated on 07/Jul/20
$${if}\:{P}_{\mathrm{2}} ^{\:{n}−{m}} =\mathrm{72}\:\:{and}\:{P}_{\mathrm{2}} ^{\:{n}+{m}} =\mathrm{210}\:{find}\:{the}\:{value}\:{of}\:{m}\:{and}\:{n}\:? \\ $$
Answered by Dwaipayan Shikari last updated on 07/Jul/20
$$\frac{\left({n}−{m}\right)!}{\left({n}−{m}−\mathrm{2}\right)!}=\mathrm{72} \\ $$$$\left({n}−{m}\right)\left({n}−{m}−\mathrm{1}\right)=\mathrm{72} \\ $$$${a}\left({a}−\mathrm{1}\right)=\mathrm{72}\:\:\:\:\left\{{take}\:{n}−{m}\:{as}\:{a}\right. \\ $$$$ \\ $$$${a}=\mathrm{9}\:\:\:\:{so}\:\:\:{n}−{m}=\mathrm{9} \\ $$$$\frac{\left({n}+{m}\right)!}{\left({n}+{m}−\mathrm{2}\right)!}=\mathrm{210} \\ $$$$\left({n}+{m}\right)\left({n}+{m}−\mathrm{1}\right)=\mathrm{210} \\ $$$${p}\left({p}−\mathrm{1}\right)=\mathrm{210}\:\:\left\{\:\:{take}\:{n}+{m}\:{as}\:{p}\right. \\ $$$$ \\ $$$${p}=\mathrm{15}\:\:\:\:{so}\:{n}+{m}=\mathrm{15} \\ $$$${so\begin{cases}{{n}=\mathrm{12}}\\{{m}=\mathrm{3}}\end{cases}} \\ $$
Commented by mohammad17 last updated on 08/Jul/20
$${thank}\:{you} \\ $$