#JavaScript (ES6), 73 65 64 bytes
Saved 1 byte by using JoKing's formula: \$1/(i+j+k+1)+1\$, with \$i\$, \$j\$, \$k\$ 0-indexed.
(x,y,z)=>(g=n=>!n||(1/(n%z-~(n/z%y)+n/z/y%x|0)+1)*g(n-1))(x*y*z)
Arnauld
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