The dependence of the luminescence quantum yield of linear polymer molecules that contain chromophores at the ends on the number of units of polymer chain is studied. Numerical calculations for the dipole-dipole energy transfer between chromophores show that an increase in the chain length by a factor of 2 leads to a several-fold increase in the luminescence quantum yield depending on the ratio of the Förster radius to the radius of polymer coil. An approximate formula is derived for the adequate analysis of the dependence at the Förster radius that is significantly less than the radius of polymer coil. The dependence of the luminescence quantum yield of linear polymer molecules on the length of the statistical segment (unit) at a constant total length of the chain is also studied.