We study shuffle product structures on words in three letters, extending the classical framework of multiple zeta values. Using an evaluation map that relates admissible words to iterated integrals, we translate shuffle identities into combinatorial identities. This yields a general multinomial-type formula and a symmetric decomposition of binomial coefficients. We also establish two new shuffle product formulas, leading to further combinatorial identities and revealing a structural connection between shuffle algebra and combinatorics.
Kwang-Wu Chen (Thu,) studied this question.