  We obtain a recursive formula for the number of rational degree $d$ curves in
$\mathbb{P}^3$, whose image lies in a $\mathbb{P}^2$, passing through $r$ lines
and $s$ points, where $r + 2s = 3d+2$. This can be viewed as a family version
of the classical question of counting rational curves in $\mathbb{P}^2$. We
verify that our numbers are consistent with those obtained by T. Laarakker,
where he studies the parallel question of counting $\delta$-nodal degree $d$
curves in $\mathbb{P}^3$ whose image lies inside a $\mathbb{P}^2$. Our numbers
give evidence to support the conjecture, that the polynomials obtained by T.
Laarakker are enumerative when $d \geq 1 + [\frac{\delta}{2}]$, which is
analogous to the {G}\"ottsche threshold for counting nodal curves in
$\mathbb{P}^2$.
