Two Einstein solids exchange four quanta with multiplicity-weighted probabilities

Two weakly coupled Einstein solids, $A$ and $B$, form an isolated composite system. Solid $A$ has $N_A=2$ distinguishable oscillators, solid $B$ has $N_B=3$, and the composite contains exactly $q_{\mathrm{tot}}=4$ indistinguishable energy quanta. The oscillator frequencies are equal, so specifying $q_A$ fixes $q_B=4-q_A$. In the Einstein-solid model, \[ \Omega(N,q)=\binom{q+N-1}{q}. \] Assume every accessible composite microstate is equally likely. 1. For each $q_A=0,1,2,3,4$, compute \[ w(q_A)=\Omega(N_A,q_A)\Omega(N_B,4-q_A). \] Give the five weights in increasing order of $q_A$ and sum them. 2. Normalize the weights to obtain $P(q_A)$. Identify the most probable macrostate and its probability. 3. Compute the exact expectation $\langle q_A\rangle$. Explain why it need not equal the modal allocation. 4. Check the total multiplicity independently by treating all five oscillators as one Einstein solid. State why multiplicities multiply for a specified split but add when mutually exclusive splits are combined. 5. State the assumptions supporting the probability model. Do not replace this microcanonical counting calculation with canonical partition-function derivatives or with the checked-in two-level fluctuation exercise.

For $N_A=2$, $\Omega_A(q_A)=q_A+1$. For $N_B=3$, use $\Omega_B(q_B)=\binom{q_B+2}{2}$.