Question

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

Answer 1

Given,

$\frac{{}^{56}{P}_{r+6}}{{}^{54}{P}_{r+3}}=30800$

$\frac{\frac{56!}{(56-r-6)!}}{\frac{54!}{(54-r-3)!}}=30800$

$\frac{56\times 55\times 54!}{(56-r-6)!}\times \frac{(54-r-3)!}{54!}=30800$

$\frac{(51-r)!}{(50-r)!}=\frac{30800}{56\times 55}$

$51-r=10$

$\therefore r=41$

Hence, this is the answer.