P0, P1 and P2 are constants. The problem is to assign a,b,c all possible values so that their sum is always 33 and compute the sum.
I suggest you start by trying to implement a simpler problem of the same type.
Sum[33! p0^a p1^b p2^c Boole[a + b + c == 33]/(a! b! c!), {a, 0, 33}, {b, 0, 33}, {c, 0, 33}]
huge output
Do you mean also b\ge0 and c\ge0? Else, for each a there are an infinite number of integer pairs a,b.
Otherwise you could generate an index set with:
t = Flatten[Table[{i, j, k}, {i, 0, 33}, {j, 0, 33}, {k, 0, 33}], 2];
indx=Select[t, (Total[#] === 33) \[And] (#[[2]] + #[[3]] > 0) &];
IntegerPartitions might be useful, but that uses positive values and needs a tweak to also include a=0
Join[Map[Join[{0}, #]&, IntegerPartitions[33, {2}]], IntegerPartitions[33, {3}]]