instead of these algebraic expressions, one can also give a combinatorial definition of gaussian binomial coefficients. the ordinary binomial coefficient formula_19 counts the -combinations chosen from an -element set. if one takes those elements to be the different character positions in a word of length , then each -combination corresponds to a word of length using an alphabet of two letters, say with copies of the letter 1 (indicating the positions in the chosen combination) and letters 0 (for the remaining positions).