Question

If ${{}^{56}P}_{r+6}:{{}^{54}P}_{r+3}=30800:1$, find $r$.

Answer 1

$\frac{{}^{56}{P}_{r+6}}{{}^{54}{P}_{r+3}}=30800⟹\frac{56!}{(50-r)!}\times \frac{(51-r)!}{54!}=30800$

$(51-r)!=10(50-r)!⟹(50-r)!\cdot (51-r)=10(50-r)!$

$⟹51-r=10⟹r=41$