Question C$1$: Joy Club produces commercial stainless steel coffee machines for various department stores. Joy Club uses an artificial intelligence system to inspect its coffee machines. This artificial intelligence system uses rigorous methods to check on coffee machines. Each coffee machine has a $75\%$ of probability to pass the inspection process.

If a coffee machine passes the inspection, it will be branded as a $$Gourmet King$$ coffee machine and be sold at a department store with a unit profit of $$355.$ If a coffee machine fails the inspection, it will be branded as an $$Easy Time$$ coffee machine and be sold on the internet with a unit profit of $$128.$ In April $2024,$ a production batch of eighteen coffee machines is collected from Joy Club. What is the probability that all eighteen coffee machines from this April $2024$ batch can be sold as $$Gourmet King$$ machines?