For a positive integer \(k\) let \(P_k(x)=x(x+1)\ldots (x+k-1)\) and \(S_k(x)=1^k+2^k+\ldots +x^k\). In the paper the following Diophantine equations are solved (or resolved): \(P_6(x)=P_4(y)\), \(P_6(x)={y\choose 2}\), \(P_6(x)={y\choose 4}\), \({x\choose 3}=P_2(y)\), \({x\choose 3}=P_4(y)\), \({x\choose 6}=P_2(y)\), \({x\choose 6}=P_4(y)\), \({x\choose 6}={y\choose 2}\), \({x\choose 6}={y\choose 4}\), \(S_2(x)=P_2(y)\), \(S_2(x)=P_4(y)\), \(S_5(x)=P_2(y)\), \(S_5(x)=P_4(y)\), \(S_5(x)={y\choose 2}\), \(S_5(x)={y\choose 4}\). The equations are reduced to elliptic equations and then the program package SIMATH is used to determine the solutions. An algorithm for finding the integer solutions of the equation \({x\choose 6}={y\choose 2}\) is illustrated, too.