Jelly, 9 bytes

Œp§’÷_ɗ2P

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Based on the closed form expression given in the question. Takes input as a list [x,y,z].

Œp           Cartesian product of the list.
             (Each element, being an integer, is implicitly converted to a range.)
  §          Sum the items of each triplet,
      ɗ      and for each sum (implicitly vectorizing)
    ÷ ɗ      divide
   ’  ɗ      the sum - 1
     _ɗ2     by the sum - 2.
        P    Return the product of all quotients.

Jelly, 9 8 bytes

Œp§_2İ‘P

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Uses the altered form of the closed form expression. Takes input as a list [x,y,z].

Œp          Cartesian product of the list.
            (Each element, being an integer, is implicitly converted to a range.)
  §         Sum the items of each triplet,
   _2       subtract 2 from each sum,
     İ      take the reciprocal of each lowered sum,
      ‘     increment each reciprocal,
       P    and return the product of the increments reciprocals.

Jelly, 9 bytes

Œp§’÷_ɗ2P

Try it online!

Based on the closed form expression given in the question.

Jelly, 9 bytes

Œp§’÷_ɗ2P

Try it online!

Based on the closed form expression given in the question. Takes input as a list [x,y,z].

Œp           Cartesian product of the list.
             (Each element, being an integer, is implicitly converted to a range.)
  §          Sum the items of each triplet,
      ɗ      and for each sum (implicitly vectorizing)
    ÷ ɗ      divide
   ’  ɗ      the sum - 1
     _ɗ2     by the sum - 2.
        P    Return the product of all quotients.