Counting patterns in colored orthogonal arrays

Let S be an orthogonal array O A ( 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 it is shown that every coloring of an orthogonal array O A ( d , d − 1 ) contains a positive proportion of almost rainbow vectors and also of almost monochromatic vectors of every color.