  Let $X$ be a non-singular quasi-projective variety over a field, and let
$\mathcal E$ be a vector bundle over $X$. Let $\mathbb G_X({d}, \mathcal E)$ be
the Grassmann bundle of $\mathcal E$ over $X$ parametrizing corank $d$
subbundles of $\mathcal E$, and denote by $\theta$ the Pl\"ucker class of
$\mathbb G_X({d}, \mathcal E)$, that is, the first Chern class of the universal
quotient bundle over $\mathbb G_X({d}, \mathcal E)$. In this short note, a
closed formula for the push-forward of powers of $\theta$ is given in terms of
the Schur polynomials in Segre classes of $\mathcal E$, which yields a degree
formula for $\mathbb G_X({d}, \mathcal E)$ with respect to $\theta$ when $X$ is
projective and $\wedge ^d \mathcal E$ is very ample.
