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 $m_{1} + m_{2} = 13$