Here the question takes wts of individual compound distribution S and T to derive distribution of U.But in getting distribution of u taking u=1,2 solution is not clear. Kindly clarify that segment.
The answer just applies the formula 3/5*P(S=x)+2/5*P(T=x). So for u=1 we have 3/5*P(S=1)+3/5*P(T=1), which gives 3/5*1/3 + 2/5*1/2. Similarly for u=2 we have 3/5*P(S=2)+3/5*P(T=2), which gives 3/5*1/3 + 2/5*1/2.