if-P-2-n-m-72-and-P-2-n-m-210-find-the-value-of-m-and-n-

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

(((n−m)!)/((n−m−2)!))=72 (n−m)(n−m−1)=72 a(a−1)=72 {take n−m as a a=9 so n−m=9 (((n+m)!)/((n+m−2)!))=210 (n+m)(n+m−1)=210 p(p−1)=210 { take n+m as p p=15 so n+m=15 so { ((n=12)),((m=3)) :}

$$\frac{\left({n}−{m}\right)!}{\left({n}−{m}−\mathrm{2}\right)!}=\mathrm{72} \\ $$$$\left({n}−{m}\right)\left({n}−{m}−\mathrm{1}\right)=\mathrm{72} \\ $$$${a}\left({a}−\mathrm{1}\right)=\mathrm{72}\:\:\:\:\left\{{take}\:{n}−{m}\:{as}\:{a}\right. \\ $$$$ \\ $$$${a}=\mathrm{9}\:\:\:\:{so}\:\:\:{n}−{m}=\mathrm{9} \\ $$$$\frac{\left({n}+{m}\right)!}{\left({n}+{m}−\mathrm{2}\right)!}=\mathrm{210} \\ $$$$\left({n}+{m}\right)\left({n}+{m}−\mathrm{1}\right)=\mathrm{210} \\ $$$${p}\left({p}−\mathrm{1}\right)=\mathrm{210}\:\:\left\{\:\:{take}\:{n}+{m}\:{as}\:{p}\right. \\ $$$$ \\ $$$${p}=\mathrm{15}\:\:\:\:{so}\:{n}+{m}=\mathrm{15} \\ $$$${so\begin{cases}{{n}=\mathrm{12}}\\{{m}=\mathrm{3}}\end{cases}} \\ $$

Commented by mohammad17 last updated on 08/Jul/20

$${thank}\:{you} \\ $$

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