Abstract For positive integers k ≤ n let P n , k ( x ) : = ∑ j = 0 k n j x j be the binomial expansion of ( 1 + x ) n truncated at the k th stage. In this paper we show the finiteness of solutions of Diophantine equations of type P n , k ( x ) = P m , l ( y ) in x , y ∈ Z under assumption of irreducibility of truncated binomial polynomials P n − 1 , k − 1 ( x ) and P m − 1 , l − 1 ( x ) . Although the irreducibility of P n , k ( x ) has been studied by several authors, in general, this problem is still open. In addition, we give some results about the possible ways to write P n , k ( x ) as a functional composition of two lower degree polynomials.