Question

Solve the following inequalities.
$C_m^{13}<C_{m+2}^{13},mϵN$

Answer 1

Ques Solve: $C_m^{13}<C_{m+2}^{13}mϵN$

By formula $=C_r^n=\frac{n!}{(n-r)!\left(r\right)!}$

We have $C_m^{13}=\frac{13!}{\left(m\right)!(13-m)!}$

and $C_{m+2}^{13}=\frac{13!}{(m+2)(13-m-2)!}$

then $C_m^{13}<C_{m+2}^{13}$

$\Rightarrow \frac{13!}{(m-13)}\Rightarrow \frac{13!}{(13-m)!\left(m\right)!}<\frac{13!}{(m+2)!(13-m-2)!}$

$\Rightarrow \frac{1}{(13-m)(13-m-1)(13-m-2)!}<\frac{1}{(m+2)!(13-m-2)!}$

$\Rightarrow \frac{1}{(13-m)(13-m-1)(m!)}<\frac{1}{\left(m\right)!(m+1)(m+2)}$

$\Rightarrow (m+1)(m+2)<(13-m)(12-m)$ by cross multiply

$\Rightarrow m^2+2m+2+2<156+m^2-25m$

$\Rightarrow 3m+25m<156-2$

$\Rightarrow 28m<154$

$\Rightarrow m<\frac{154}{28}$

$m<\frac{77}{14}$

So $m<\frac{77}{14}$