In a radioactive element the fraction of initial amount remaining after its mean life time is :

The correct option is C $\frac{1}{e}$
$A_0$ = initial number of radioactive nuclei
A = radioactive nuclei at $t=t$
$A=A_0{e}^{-\lambda t}$
$t_{\text{mean}}=\frac{1}{\lambda }\Rightarrow t_{\text{mean}}\lambda =1$
$A=A_0{e}^{-1}$
$\frac{A}{A_0}=e^{-1}=\frac{1}{e}$