Berachos, Chapter 8, Mishnah 4. Here we have another dispute between Beis Shammai and Beis Hillel concerning the meal, except that this time the Mishnah deals with the conclusion of the meal. The subject is mayim acharonim - the washing we do at the end of the meal, before Birkas HaMazon - and the question is which comes first: washing mayim acharonim, or clearing the crumbs off the table?
The words of the Mishnah:
"Beis Shammai omrim: mechabdin es habayis ve'achar kach nosnin la'yadayim" - first the crumbs are cleared off the table, and only afterwards does one wash mayim acharonim.
"Beis Hillel omrim: nosnin la'yadayim ve'achar kach mechabdin es habayis" - first one washes mayim acharonim, and afterwards the table is cleared.
The basis of the dispute:
According to the beraisa and the Gemara, the root of the dispute lies in the question of whether one may make use of a waiter (the attendant who assists at the meal) who is an unlearned person and is not well versed in halachah:
The view of Beis Shammai: Since one may make use of a waiter who is not well versed in halachah, there is reason to be concerned that he will not understand that he must clear off the table, before the end of the meal, those pieces of bread that are the size of a kezayis. And since crumbs the size of a kezayis are liable to remain on the table, if a person washes his hands first, the moisture on his hands may ruin the crumbs and render them unfit to eat. Therefore one should first clear the room and remove the large pieces that the waiter left behind, and only afterwards wash mayim acharonim.
The view of Beis Hillel: Since one may make use only of a waiter who is a talmid chacham, who knows the halachah and knows that it is forbidden to leave over pieces the size of a kezayis, he removes them beforehand, even before the washing. Therefore one may wash mayim acharonim first and clear the room afterwards, for even if the crumbs do get wet, they will never be anything more than less than a kezayis.
In summary: In this Mishnah, Beis Shammai and Beis Hillel disagree about the order of precedence at the conclusion of the meal: according to Beis Shammai one clears the room and afterwards washes the hands, and according to Beis Hillel one washes the hands and afterwards clears the room. The basis of the dispute is the question of using a waiter who is an unlearned person: Beis Shammai were concerned that he would leave over pieces the size of a kezayis that might be ruined by the moisture of the washing, while Beis Hillel held that the waiter is a talmid chacham, and he will leave nothing on the table but less than a kezayis.