Consider a portfolio made of two independent classes, with $n_1$ and $n_2,1-$year term life insurance contracts,

respectively. The contracts are issued to policyholders with independent and identically distributed $($i$.$i$.$d$)$

future lifetimes $T_1,..,T_n.$ The death benefit amount for all the contracts is the $b_i=b,i=1,$dots,$n.$

Moreover, for the contracts of the first class we have:

$Pr(T_{i}1)=q_1,i=1,$dots,$n_1,$

while for the contracts of the second class, we have:

$Pr(T_{i}1)=q_2,i=n_1+1,$dots,$n_1+n_2.$

The costs for each contract i are defined according to the indemnity approach.

Find an explicit expression for $Pr(S=kb),$ where $S$ is the total amount of claims for the whole portfolio.