  Let $S$ be an orthogonal array $OA(d,k)$ and let $c$ be an $r$--coloring of
its ground set $X$. We give a combinatorial identity which relates the number
of vectors in $S$ with given color patterns under $c$ with the cardinalities of
the color classes. Several applications of the identity are considered. Among
them, we show that every equitable $r$--coloring of the integer interval
$[1,n]$ has at least $1/2(n/r)^2+O(n)$ monochromatic Schur triples. We also
show that in an orthogonal array $OA(d,d-1)$, the number of monochromatic
vectors of each color depends only on the number of vectors which miss that
color and the cardinality of the color class.
