The prohibition of "Amira Leakum" forbids asking a non-Jew on Shabbat to perform an activity which is forbidden for a Jew to perform on Shabbat. However, it is permissible to ask a non-Jew on Shabbat to do a task which can be done in a permissible way, even if one knows that the non-Jew will do it in a manner which is forbidden for a Jew. In such a case, the non-Jew performs an act which is forbidden for a Jew on Shabbat, only for his own convenience; the Jew did not ask him to perform that act. As such, this is permissible.
One example would be ordering from a non-Jewish waiter in a hotel on Shabbat. Even though the guest sees the waiter with a pen and paper in hand, and he knows the waiter will write down his order, it is nevertheless permissible to place the order. Since it is possible for the waiter to commit the order to memory, rather than write it down, the Jew in this case does not ask the waiter to perform an act which is forbidden on Shabbat. The waiter writes only for his own convenience, because this is easier than remembering the order. Therefore, since the Jew does not ask the non-Jew to perform an act that is forbidden on Shabbat, this is permissible. (It would be forbidden, however, to specifically tell the waiter to write down the order to ensure he does not forget it.)
Likewise, it is permissible to ask a non-Jew to bring something from a room on a high floor in a building, even though one knows that the non-Jew will take the elevator. Since it is possible for the non-Jew to take the stairs, and he takes the elevator only for his own convenience, this is permissible. Another example is asking a non-Jewish housekeeper to wash the dishes on Friday night so they will be ready Shabbat morning. The housekeeper can wash them by hand, and so even if the Jew knows that the housekeeper will place them in the dishwasher and turn it on, this is permissible. The same applies if the housekeeper will wash the dishes by hand with hot water. Since the Jew does not ask her to turn on the hot water, this is permissible. Likewise, one may ask a non-Jewish housekeeper to wipe the table even if the housekeeper will use a wet cloth and perform an act of Sehita (squeezing), which is forbidden on Shabbat. Since the table does not need to be cleaned in this fashion, such a request may be made.
Another application of this Halacha is asking a non-Jewish custodian to clean the carpet in the synagogue after services. Since it is possible for the custodian to clean the carpet by hand, or with a broom, he may be asked to clean even if it is known that he will use a vacuum cleaner. Although it would be inappropriate for him to run the vacuum cleaner while the congregants are still present in the synagogue, it is permissible to ask him to clean after they leave, even though he will use a vacuum cleaner.
Summary: It is permissible to ask a non-Jew on Shabbat to do a task which can be done in a manner which is permissible for Jews, even if it is clear that the non-Jew will do the task in a manner which is forbidden for Jews. For example, a hotel guest may place an order on Shabbat with a non-Jewish waiter even though he will definitely write down the order, and one may ask a non-Jewish housekeeper on Shabbat to wash the dishes even if she will definitely use the dishwasher.
