Convert a sample to an index
Jelly, 8 bytes

S0rṗLSÞi


Try it online!

Brute-force solution. The first test case is too much for TIO, but I've verified it locally on my laptop. The second test case requires too much RAM, even for my desktop computer.

How it works

S0rṗLSÞi  Main link. Argument: A (array)

S         Compute the sum s of A.
 0r       Create the range [0, ..., s].
    L     Yield the length l of A.
   ṗ      Cartesian power; yield the array of all l-tuples over [0, ..., s], in
          lexicographical order.
     SÞ   Sort the l-tuples by their sums. The sorting mechanism is stable, so
          l-tuples with the same sum are still ordered lexicographically.
       i  Find the index of A in the generated array of tuples.