This is a homework assignment, can someone provide some guidance (not a full solution):
There are three types of blocks: $A$, $B$, and $C$. All blocks look exactly the same, but they have a different weight: $w_A$, $w_B$, and $w_C$ respectively.
I have a total $n_T$ of blocks, but I don't know the amount of each block; I only know the total weight $w_T$. Use maximum entropy to obtain the joint probability.
My Attempt
I want to obtain $P(n_A, n_B, n_C)$ given the following constraints: $$ n_A + n_B + n_C = n_T $$ $$ w_An_A + w_Bn_B + w_Cn_C = w_T$$ $$ \int_0^{n_T}\int_0^{n_T}\int_0^{n_T} P(n_A, n_B, n_C) \text{d}n_A\text{d}n_B\text{d}n_C = 1$$
I form the lagrangian function: $$\mathcal{L} = -\int P(n_A, n_B, n_C)\ln P(n_A, n_B, n_C)\text{d}^3n - \lambda \left[\int P(n_A, n_B, n_C)\text{d}^3n - 1 \right] - \lambda_1\left[n_A + n_B + n_C - n_T\right] - \lambda_2\left[w_An_A + w_Bn_B + w_Cn_C - w_T\right]$$
But when I do the derivative w.r.t. $P$ I just end up with $$P(n_A, n_B, n_C) = e^{-\lambda - 1}$$ And I think this is just telling be that my joint distribution is uniform. I don't think I am applying my constraints correctly. Does anybody have any suggestions?