Gemara

(Mishnah 1): These laws (above) are if the Kohen asked. If he did not ask, and each of Rachel and Leah gave one Chovah, or each gave two, or three:

If he offered all of them above, half of them are valid (the Olos);

If he offered all below, half are valid (Chata'os);

If he offered half of them above and half below, half of those above are valid, and half are Pesulim.

The same applies to those offered below.

(Mishnah #2): If one woman gave one Ken, another gave two, another gave three, another gave ten, and another gave 100, and all were offered above, half of them are valid;

If all were offered below, half are valid;

If he offered half the birds above and half below, the largest number (of Kinim) of any woman (100) are Kosher.

The general rule is, if it is possible that no woman had birds offered both above and below, half are valid;

If we must say that a woman had birds offered both above and below, the number of valid Kinim is the greatest number of Chovos of any woman.

Zevachim 73b (Rava): Since Chachamim forbade offering Korbanos (that became mixed, they are Nidchim, i.e. permanently disqualified;) if any were offered, they are Pesulim.

Question (Rav Huna bar Yehudah - Mishnah): If Chata'os and Olos ha'Of became mixed together, even if one was mixed with 10,000 of the other, all must be left to die;

If a Kohen did not ask and offered all of them above, half are Kesherim. If he offered all of them below, half are Kesherim. (According to Rava, all should be Pesulim!)

Answer: Rava's law is like the opinion that a living being can be Nidcheh. The Mishnah is like the opinion that it cannot.

Kerisus 7b (Beraisa): If two women miscarried, and only one of the fetuses obligates a Korban, and it is not known whose, they bring Korbanos and they are not eaten;

R. Yosi says, they bring a Korban and it is eaten;

They bring one Chatas ha'Of together and stipulate (that it is for the one who is obligated).

Rishonim

Rambam (Hilchos Pesulei ha'Mukdashim 8:6): If two Chovos Stumos were mixed, one greater than the other, e.g. one was four birds and one was six, if he offered all of them above, or all below, half are Pasul and half are valid. If he offered half of them above and half below, if he did so after asking, the smaller number are Kosher. If he did so on his own, the larger number are Kosher.

Rebuttal (Ra'avad): The Rambam learned from the Mishnah 'this refers to a Kohen who asked. If he did not ask, if each woman gave... half are Kosher. ' The Rambam's words are messed up. If he offered all of them above, or all below, it does not matter whether or not he asked! In any case half are Pasul and half are valid. Rather, if he asked, we tell him to offer the smaller number of Kinim, half above and half below. We do not (intentionally) offer Pesulim on the Mizbe'ach. He may not offer another Chatas, for perhaps it is from the (woman who had) the smaller number, and he already offered the full number of Chata'os. Likewise, he may not offer another Olah. When the Kohen did not ask and offered half above and half below, the larger number are Kosher. Do not explain that the woman with more Kinim fulfilled her obligation. She does not know how many of hers were offered above and below. Rather, she brings with the other woman (the remainder, i.e. the smaller number) and stipulates. Whatever was offered properly, decreases the amount they must bring afterwards.

Ri Korkus: Perush ha'Mishnayos says that our Mishnah teaches when it matters whether or not the Kohen asked. If the two women have the same number of Kinim, whether or not he asked, whether he offered all above, or all below, or half above and half below, half are Kosher. However, if Rachel has one Kan and Leah has two, if he offered all above, or all below, in any case half are Kosher. However, if he offered half above and half below, if he asked, the smaller number are Kosher. If he did not ask, the larger number are Kosher. Perush ha'Mishnayos explains that 'Nimlach' (he asked) means 'he needed to ask' (permission), because they had different numbers. In the Mishneh Torah, he explains that it means that he asked.

Ri Korkus: According to the Ra'avad, also when the Kohen offered all above, the number of Kosher Olos is the larger number of Kinim.

Note: I do not understand this. Only half of them can be Olos!

Ri Korkus: The Ra'avad's law is correct, but it is difficult to say that the Mishnayos, which are brought also in Zevachim 67b, discuss a Kohen who erred and did not ask. The Tana should have specified! The Mishnah connotes that he may offer all of them. The Mishnah says 'half and Kosher and half are Pasul', i.e. half may be offered, and half may not be offered. Similarly, 'the smaller number are Kosher' permits offering the smaller number. In Perek 3, the Mishnah discusses one who offered all of them, so Kosher and Pasul refer to what counted for their obligations. The Rambam holds that always, one may offer all of them. The Mefaresh explains like the Ra'avad. The Rambam did not explain why it matters whether or not the Kohen asked.

Kesef Mishneh: The Mefaresh says that our Mishnah refers to the first Perek. If an Olah that became mixed with Chata'os, or vice-versa, all must die, or if Chovos of 1, 2 and 3 Kinim became mixed..., That is l'Chatchilah, when the Kohen asked what to do. If the Kohen did not ask, and offered a whole Kan above for one woman, and a whole Kan (below) for another woman, half are Kesherim. Also Zevachim 73b connotes that our Mishnah refers to the first Perek. Perush ha'Mishnayos says that our Mishnah refers to Mishnah 1:4, which says that the smaller number is Kosher. If the Kohen did not ask, and offered half above and half below, the larger number are Kosher. I do not understand his distinction.

Rama mi'Piano (23): When birds became mixed with dissimilar groups (e.g. Chata'os in Chovos), the Rambam holds that 'are Kosher only...' refers to what one may offer l'Chatchilah. He says (in Perush ha'Mishnayos) that the third Perek refers to Mishnah 1:4, which teaches what is Kosher b'Di'eved. Surely Perek 2 discusses Kashrus b'Di'eved, for the Tana did not say 'the amount that may be offered.' He holds that it depends on asking the women, and not asking Beis Din. In Zevachim clearly it refers to asking Beis Din. The Rambam holds that that is a Beraisa. It relies on the Reisha, which says that all must die. (So clearly, it refers to asking Beis Din - PF.)

Rama mi'Piano: According to the Rambam, the Mishnah praises the Kohen for offering in a way to be Machshir as many as possible. This answers the questions of the Ra'avad and Kesef Mishneh. However, surely we discuss asking Beis Din. If it was initially Mefureshes (specified), all must die (Mishnah 2:4). If (it was Mefureshes) at the end, why is the woman better than the Kohen (to require asking her)? They were mixed before they were offered! The Rambam can say that b'Di'eved, if a Mefureshes was offered, some of them are Kosher. Even if we discuss asking Beis Din, we praise a Kohen who did not ask, just like a zealot (such as Pinchas) may kill (certain) transgressors only if he did not ask. We hold that Kohanim are Sheluchim of Hash-m. If a Kohen asks the owner, the owner relies on the Kohen only for the smaller amount. If the Kohen did not ask the owner, he is Hash-m's Shali'ach. Hash-m has no doubt, therefore the larger number are Kosher.

Merkeves ha'Mishneh (8:3 DH veha'Nachon): The Halachah does not follow R. Yosi, who says that two may stipulate about Chatas ha'Of. Therefore, if we do not know whose Kan was Kosher, both must bring another Kan! Therefore, the Rambam learns unlike the other Meforshim. Rather, the women bought the birds together and did not specify which birds are for whom. They are not determined until the Kohen offers them. They gave to him individual Kinim, e.g. AB, CD, EF, GH and IJ. He should have consulted them. They might have told him to offer ACEGI above and BDFHJ below, or vice-versa. If the Kohen offered half, e.g. ABCDE above, at least two are Kosher (A or B, and C or D), for one of each Kan needed to be offered above. Likewise, of FGHIJ offered below, at least two are Kosher. However, perhaps the women would have told him to offer E below and F above. It is as if he offered these without permission, so they are both Pasul. (Had the Kohen offered ACEGI above, we would not say that perhaps all are Pesulim, for it is a Zechus for both women that they be Kosher, and Zachin l'Adam she'Lo Befanav.) If Rachel gave two Kinim and Leah gave three, we could say that all Rachel's Kinim are the two Kosher ones, but we cannot say so about Leah. Therefore, we attribute both Kosher Kinim to Rachel.

Note: It seems that it is like a Tanai of Beis Din to attribute as many Kosher Kinim as we can to the woman in a 'better' situation, e.g. we know that she has at least one Kosher Kan. If neither is guaranteed to have a Kosher Kan, if it is possible that all one woman's Kinim are Kesherim, her situation is 'better' - PF.

Merkeves ha'Mishneh: If the Kohen did not ask them, he offered with permission, so also the Kan EF is Kosher. Surely Leah has at least one Kosher Kan; perhaps Rachel has none. Therefore, we attribute all three Kosher Kinim to Leah, and none to Rachel. Likewise, if Rachel gave 10 and Leah gave 100, and the Kohen did not ask and offered 110 above and 110 below, surely Leah has some of the 55 Kosher Kinim, but perhaps Rachel has none, so all 55 count for Leah. If the Kohen asked, he acted without permission. Rachel can say 'perhaps all mine are among the Kosher Kinim.' Leah cannot say so. Therefore we say that all 10 Kinim of Rachel were Kosher, and only 45 of Leah were Kosher.

Merkeves ha'Mishneh (6): Here, the Rambam retracted from Perush ha'Mishnayos and explained like the Ra'avad. The Kinim were Mefurashes; it depends on whether or not he asked Beis Din. After he asked, even b'Di'eved it (the excess over the smaller number) is not acceptable. Therefore, the Kosher Kinim are of Rachel (who gave less), like I explained above. If he did not ask, Leah's Kinim are Kosher. Rachel Vadai must offer all her Kinim again. How can the Ra'avad say that they bring the rest with a Tanai? R. Yosi permits bringing Chatas ha'Of on Tanai, but the Halachah does not follow him!

Perush of father of Yair Kino (at the end the Mishnayos, p.54, DH Lachen): The Rambam means that if the Kohen did not ask, Leah (who gave more Kinim) surely has Kosher Kinim, therefore we attribute all the Kosher Kinim to her. If the Kohen asked, and the women specified two birds for each Ken, and afterwards they became mixed, perhaps only the smaller amount is Kosher. Therefore, we attribute them to Rachel.