  We prove that some Gromov-Witten numbers associated to rational contact
(Legendrian) curves in any contact complex projective space with arbitrary
incidence conditions are enumerative. Also, we use Bott formula on the
Kontsevich space to find the exact value of those numbers. As an example, the
numbers of rational contact curves of low degree in $\mathbb{P}^{3}$ and
$\mathbb{P}^{5}$ are computed. The results are consistent with existing
results.
