A pair of students is selected at random from a probability class. The probability that the pair selected will consist of one male and one female student is $\frac{10}{19}.$  If the maximum number of students the class can contain is $n,$ find the number of positive divisors of  $n.$

Let  $t=m+w.$
We are given that  $\frac{10}{19}=\frac{mw}{{ }^t{C}_2}=\frac{2mw}{t(t-1)}\mathrm{............}\left(1\right)$
But  $\sqrt{mw}\le \frac{m+w}{2}or2\sqrt{mw}\le t(G.M.\le A.M.)$
or  $mw\le \frac{t^2}{4}\mathrm{..........}\left(2\right)$
Substituting  $mw=\left(\frac{10}{19}\right)\left(\frac{t(t-1)}{2}\right)$
We have  $\left(\frac{5}{19}\right)t(t-1)\le \frac{t^2}{4}\Rightarrow 20(t^2-t)\le 19t^2$
$\Rightarrow 20t^2-20t\le 19t^{2}or{t}^2\le 20t\Rightarrow t\le 20$
 $\Rightarrow t_{\max }=20$
Also equality holds if  $m=w=10$
$\therefore$    Maximum children in class = 20