Discuss the method of lagrangian multipliers

Question

Response to the following problem:

Given N indistinguishable, quasi-independent particles capable of existing in energy levels ?₁, ?₂,· .. , with degeneracies g_b g₂, ... , respectively; in any given macro state in which there are N, particles in energy level ?₁, N₂ particles in energy level ?₂, ... , assume the thermodynamic probability to be given by the Bose-Einstein expression,

Ω_BE = (g₁ + N₁)!(g₂ +N₂)!.../g₁!N₁!g₂!N₂!

Using Stirling's approximation and the method of Lagrangian multipliers, render

In Ω_BE a maximum,subject to the equations of constraint ∑N_i = N = const.and ∑N_i?_i = U = const., and show that

                             N_i = g_i/λe^{-β?_i - 1}