Question

If ${}^n{}^{+}{}^1{P}_5:{}^{n}P{}_6=$ $2:7$, then $n$$=$

• A. $8$
• B. $9$
• C. $10$
• D. $11$

Answer 1

The correct option is D $11$

From the question, we have

$\frac{{}^{n+1}{P}_5}{{}^n{P}_6}=\frac{2}{7}$

$\Rightarrow \frac{\frac{(n+1)!}{(n-4)!}}{\frac{n!}{(n-6)!}}=\frac{2}{7}$

$\Rightarrow \frac{(n+1)}{(n-4)(n-5)}=\frac{2}{7}$

$\Rightarrow 7n+7=2(n^2-9n+20)$
$\Rightarrow 2n^2-25n+33=0$
$\Rightarrow 2n^2-25n-3n+33=0$
$\Rightarrow (2n-3)(n-11)=0$
$\Rightarrow n=\frac{3}{2}$ or $11$
$\Rightarrow n=11$