Under very mild conditions, we prove that the number of components in a dec
omposable logarithmic combinatorial structure has a distribution which is c
lose to Poisson in total variation. The conditions are satisfied for all as
semblies, multisets and selections in the logarithmic class. The error in t
he Poisson approximation is shown under marginally more restrictive conditi
ons to be of exact order O(1/ log n), by exhibiting the penultimate asympto
tic approximation; similar results have previously been obtained by Hwang [
20], under stronger assumptions. Our method is entirely probabilistic, and
the conditions can readily be verified in practice.