Question

Solve the following inequalities.
$C_{m-2}^{13}>C_m^{18},mϵN$

Answer 1

$C_{m-2}^{18}>C_m^{18}>C_m^{18}m$

Using the formula $C_r^n=\frac{n!}{(n-r)!r!}$

We have

$C_{m-2}^{18}>C_m^{18}$

$\Rightarrow \frac{\left(18\right)!}{(m-2)!(18-m+2)!}>\frac{\left(18\right)!}{\left(m\right)!(18-m)!}$

$\Rightarrow \frac{1}{(m-2)!(18-m+1)(18-m+2)(18-m)!}>\frac{1}{\left(m\right)(m-1)(m-2)!}$

$\Rightarrow \frac{1}{(18+1-m)(18-m+2)}>\frac{1}{m(m-1)}$

$\Rightarrow m(m-1)>(19-m)(20-m)$ (Cross multiply)

$\Rightarrow m^2-m>(380+m^2-39m)$ as $m^2>0$

$\Rightarrow -m>380-39m$

$\Rightarrow 39m-m>380$

$\Rightarrow 38m>380$

$\Rightarrow m>\frac{380}{380}$

$\Rightarrow m>10$