Chapter 5, Mishnah 3. The Mishnah opens with a list of people for whose sake, if one slaughters the Pesach offering intending that they eat from it, the offering is disqualified. The reason: regarding the Pesach offering the Torah says "ish lefi ochlo" - each person according to what he eats, meaning that every person is registered for it according to his ability to eat from it. Therefore, one who slaughters with the intention of feeding people who cannot eat from the Pesach - his offering is not valid.
The disqualifying intentions at the time of slaughter:
"Shechato shelo l'ochlav" - he slaughtered it for the sake of someone who cannot eat from it, such as a sick person or an elderly person.
"V'shelo limnuyav" - for the sake of someone who is not registered for this particular Pesach offering. A person must register in advance for a specific Pesach; and if three people registered together for the offering, and the one slaughtering has in mind a fourth person who did not register in advance, the Pesach is not valid, since that fourth person cannot eat from this offering.
"La'areilim" - he slaughtered it for the sake of someone who has not had a bris milah. An areil, even though he is a Jew, cannot eat from the Pesach offering.
"V'litmei'im" - he slaughtered it for the sake of someone who is tamei, who cannot eat from the Pesach because of his tumah.
All of these are invalid.
Mixed intention:
What is the law when the intention is mixed, both for those who are fit to eat and for those who are not?
"L'ochlav v'shelo l'ochlav" - for those who can eat from the Pesach and for those who cannot.
"Limnuyav v'shelo limnuyav" - for those who are registered for this Pesach offering and for those who are not.
"L'mulim va'areilim" - for those who have had a bris milah and for those who have not.
"Litmei'im ut'horim" - for those who are tamei and for those who are tahor.
In all of these cases the offering is valid, because at the very least the intention includes someone who can eat from the Pesach.
Although in the previous Mishnah we learned that a mixed intention of "lishmo v'shelo lishmo" - for the sake of the Pesach and for the sake of another offering - does disqualify, the two cases are not comparable. There the intention relates to the essence of the Pesach itself, for he is slaughtering the Pesach offering itself not for its own sake, and therefore even a partial admixture of shelo lishmo intention is enough to disqualify. Here, however, the intention relates to something outside the offering, to the people who will eat from it, and therefore a mixed intention does not disqualify.
The time of the slaughter:
The Mishnah continues: "Shechato kodem chatzos - pasul, mishum shene'emar bein ha'arbayim" - if he slaughtered it before midday it is invalid, because the Torah says "in the afternoon." One who slaughters the Pesach before midday on Erev Pesach - it is invalid, since the Torah states that it must be slaughtered in the afternoon, that is, from the sixth hour of the day onward. In practice we do not slaughter until the middle of the seventh hour, when it is already clear that the afternoon has arrived.
As we mentioned at the beginning of the chapter, the Pesach offering is unique in that it is slaughtered after the afternoon tamid offering. Nevertheless, if it was slaughtered before the tamid, even though this is not the proper order, it is valid, provided that someone stirs its blood until after the blood of the tamid has been thrown, and only then may he throw the blood of the Pesach. The reason for the stirring is so that the blood should not congeal, so that it remains fit to be thrown on the altar.
And even if the blood of the Pesach offering was thrown on the altar before the blood of the tamid, although one should not act this way, it is valid; it is simply that we are careful from the outset that the blood of the tamid should come first.