  We study pairs and m--tuples of compositions of a positive integer n with
parts restricted to a subset P of positive integers. We obtain some exact
enumeration results for the number of tuples of such compositions having the
same number of parts. Under the uniform probability model, we obtain the
asymptotics for the probability that two or, more generally, m randomly and
independently chosen compositions of n have the same number of parts. For a
large class of compositions, we show how a nice interplay between complex
analysis and probability theory allows to get full asymptotics for this
probability. Our results extend an earlier work of B\'ona and Knopfmacher.
While we restrict our attention to compositions, our approach is also of
interest for tuples of other combinatorial structures having the same number of
parts.
