  We study the two dimensional three-body problem in the general case of three
distinguishable particles interacting through zero-range potentials. The
Faddeev decomposition is used to write the momentum-space wave function. We
show that the large-momentum asymptotic spectator function has the same
functional form as derived previously for three identical particles. We derive
analytic relations between the three different Faddeev components for three
distinguishable particles. We investigate the one-body momentum distributions
both analytically and numerically and analyze the tail of the distributions to
obtain two- and three-body contact parameters. We specialize from the general
cases to examples of two identical, interacting or non-interacting, particles.
We find that the two-body contact parameter is not a universal constant in the
general case and show that the universality is recovered when a subsystem is
composed of two identical non-interacting particles. We also show that the
three-body contact parameter is negligible in the case of one non-interacting
subsystem compared to the situation where all subsystem are bound. As example,
we present results for mixtures of Lithium with two Cesium or two Potassium
atoms, which are systems of current experimental interest.
