Question

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

• A. $40$
• B. $41$
• C. $42$
• D. $39$

Answer 1

The correct option is B $41$

Note that ${}^{56}{P}_{r+6}=\frac{56!}{\left(56-r-6\right)!}=\frac{56!}{\left(50-r\right)!}$ and ${}^{54}{P}_{r+3}=\frac{54!}{\left(54-r-3\right)!}=\frac{54!}{\left(51-r\right)!}$
Since ${}^{56}{P}_{r+6}:{}^{54}{P}_{r+3}=30800:1$, we have

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

$\Rightarrow \frac{56!}{\left(50-r\right)!}\times \frac{\left(51-r\right)!}{54!}=30800$

$\Rightarrow 56\times 55\times \left(51-r\right)=30800$
$\Rightarrow 51-r=\frac{30800}{56\times 55}=10$
$\therefore r=41$