Abstract: We present a multinomial theorem for elliptic commuting variables. This result extends the speaker's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a simple elliptic star-triangle relation, ensuring the uniqueness of the normal form coefficients, and, for the recursion of the closed form elliptic multinomial coefficients, the Weierstraß type A elliptic partial fraction decomposition. From our elliptic multinomial theorem we obtain, by convolution, an identity that is equivalent to Rosengren's type A extension of the Frenkel-Turaev 10-V-9 summation, which in the trigonometric or basic limiting case reduces to Milne's type A extension of the Jackson 8-phi-7 summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice Z^r, our derivation of the A_r Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity, and, at the same time, of important special cases including the A_r Jackson summation.