In this paper, we study the sign-changing radial solutions of the following coupled Schrödinger system{−Δuj+λjuj=μjuj3+∑i≠jβijui2ujin B1,uj∈H0,r1(B1) for j=1,⋯,N. Here, λj,μj>0 and βij=βji are constants for i,j=1,⋯,N and i≠j. B1 denotes the unit ball in the Euclidean space R3 centred at the origin. For any P1,⋯,PN∈N, we prove the uniqueness of the radial solution (u1,⋯,uj) with uj changes its sign exactly Pj times for any j=1,⋯,N in the following case: λj≥1 and |βij| are small for i,j=1,⋯,N and i≠j. New Liouville-type theorems and boundedness results are established for this purpose.