Question

If $m_1$ and $m_2$ satisfy the relation ${}^{m+5}{P}_{m+1}=\frac{11}{2}\left(m-1\right)\left({}^{m+3}{P}_m\right)$, then $m_1+m_2$ is equal to

• A. $10$
• B. $9$
• C. $13$
• D. $17$

Answer 1

Answer: (C)

Let ${}^{m+5}{P}_{m+1}=\frac{11}{2}\left(m-1\right)\left({}^{m+3}{P}_m\right)$
$\Rightarrow \frac{\left(m+5\right)!}{4!}=\frac{11}{2}\left(m-1\right)\left[\frac{\left(m+3\right)!}{3!}\right]$
$\Rightarrow \frac{\left(m+5\right)\left(m+4\right)}{4}=\frac{11}{2}\left(m-1\right)$
$\left(m+4\right)\left(m+5\right)=22\left(m-1\right)$
$\Rightarrow m^2-13m+42=0$
$\Rightarrow m^2-7m-6m+42=0$
$\Rightarrow m\left(m-7\right)-6\left(m-7\right)=0$
$\Rightarrow m=6,7$
Hence $m_1+m_2=13$