  In this paper we use the Lyndon-Shirshov basis to study the shuffle type
polynomials. We give a free noncommutative binomial (or multinomial) theorem in
terms of the Lyndon-Shirshov basis. Another noncommutative binomial theorem
given by the shuffle type polynomials with respect to an adjoint derivation is
established. As a result, the Bell differential polynomials and the $q$-Bell
differential polynomials can be derived from the second binomial theorem. The
relation between the shuffle type polynomials and the Bell differential
polynomials is established. Finally, we give some applications of the free
noncommutative binomial theorem including application of the shuffle type
polynomials to bialgebras and Hopf algebras.
