An approach to the stable solution of discrete ill-posed problems, based on a combination of random projection of the initial ill-conditioned matrix with ill-defined numerical rank and pseudo-inversion of the resulting matrix. To select the dimension of the projection matrix we propose to use model selection criteria. The results of experimental studies on the well-known examples of discrete ill-posed problems are given. Solution error is close to the Tikhonov regularization error, however the matrix dimension reduction due to the projection provides the possibility to decrease computational costs, especially at high noise levels.