  In the paper we develop an approach to asymptotic normality through factorial
cumulants. Factorial cumulants arise in the same manner from factorial moments,
as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is
a new identity for "moments" of partitions of numbers. The general limiting
result is then used to (re-)derive asymptotic normality for several models
including classical discrete distributions, occupancy problems in some
generalized allocation schemes and two models related to negative multinomial
distribution.
