#JavaScript (ES6), <s>&nbsp;73 65&nbsp;</s> 64 bytes

Saved 1 byte by using [JoKing's formula][1]: \$1/(i+j+k+1)+1\$, with \$i\$, \$j\$, \$k\$ 0-indexed.

<!-- language-all: lang-javascript -->

    (x,y,z)=>(g=n=>!n||(1/(n%z-~(n/z%y)+n/z/y%x|0)+1)*g(n-1))(x*y*z)

[Try it online!](https://tio.run/##bc3BDoIwEATQu1@BB5JdKJYtePBQ/oUgEI3ZGjGGNsRfrz0hATKHObxk5l5/6qF53Z7vjM219Z32MAorHOoKes26OvI0AUng2GVfYOlii2koaeNxyjElTHrgjBBhTGzi0DeGB/NoTw/TQwckQhAjKSNdReqwVTVrsVElilkpX3HAxfQulzOXl7UHXXzveSHO//mSSPkf "JavaScript (Node.js) – Try It Online")


  [1]: https://codegolf.stackexchange.com/a/196074/58563