Commentary for Kinnim 3:5
חַטָּאת שֶׁנִּתְעָרְבָה בְחוֹבָה, אֵין כָּשֵׁר אֶלָּא מִנְיַן חַטָּאת שֶׁבַּחוֹבָה. חוֹבָה שְׁנַיִם בְּחַטָּאת, מֶחֱצָה כָשֵׁר וּמֶחֱצָה פָסוּל. וְחַטָּאת שְׁנַיִם בְּחוֹבָה, הַמִּנְיָן שֶׁבַּחוֹבָה כָּשֵׁר. וְכֵן עוֹלָה שֶׁנִּתְעָרְבָה בְחוֹבָה, אֵין כָּשֵׁר אֶלָּא מִנְיַן עוֹלוֹת שֶׁבַּחוֹבָה. חוֹבָה שְׁנַיִם בְּעוֹלָה, מֶחֱצָה כָשֵׁר וּמֶחֱצָה פָסוּל. עוֹלָה שְׁנַיִם בְּחוֹבָה, הַמִּנְיָן שֶׁבַּחוֹבָה כָּשֵׁר:
Sin-offering birds that got mixed with an [equal number of] obligatory pairs [of sin-offerings and burnt offering] they are not valid, except for the number of sin-offering birds within the obligatory pairs. If the number of obligatory pairs are twice as many as the sin-offering birds, half are valid and half are invalid. If the sin-offering birds are twice as many as the obligatory [pairs] only the number of birds in the obligatory pairs are valid. Similarly, a burnt-offering that got mixed with an obligatory pair only the number of burnt-offering birds in the obligatory pairs are valid. If the number of obligatory [pairs] are twice as many as the bunt-offering [birds], half are valid and half are invalid. If the burnt-offering [birds] are twice as many as the obligatory [birds] only the number [of birds] in the obligatory pairs are valid.
Bartenura on Mishnah Kinnim
English Explanation of Mishnah Kinnim
If the [unassigned] obligatory offerings are twice as many as the hataot, then half are valid and half invalid;
But if the hataot are twice as many as the [unassigned] obligatory offerings, then the number [of hataot] among the obligatory offerings are valid.
So, too, if [birds assigned as] olot were mixed up with [unassigned] obligatory offerings, only the number of olot among the obligatory offerings are valid.
If the [unassigned] obligatory offerings are twice as many as the olot, then half are valid and half invalid.
But if the olot are twice as many as the [unassigned] obligatory offerings, then the number [of olot] among the obligatory offerings are valid.
Section one: This halakhah is basically a repeat of the halakhah taught in 1:2. Obligatory bird offerings contain two birds one of which will be an olah and one a hatat. So if some birds that have already been designated as hataot get mixed up with some pairs of obligatory offerings, the valid birds are equivalent to the number of hataot in the pairs. So if ten hataot get mixed up with six pairs, of the 22 birds, any six we take can definitely be a hatat. But the seventh bird might be the seventh bird from the six pairs, in which case it should have been an olah.
Section two: If two obligatory offerings, meaning two birds one of which needs to be a hatat and one an olah, are mixed up with one hatat, then we divide the birds into two. The first bird is valid as a hatat, the second bird is definitely disqualified and the third bird is potentially half-valid, but since it can't be divided, there is nothing that can be done with it. Similarly, if three obligatory offerings get mixed up with one hatat, three birds are valid as a hatat, three are invalid and the third cannot be divided.
Section three: But if there are two hataot that get mixed up with one pair of obligatory offerings, only one bird will be valid as a hatat.
Sections four-six: The second half of this mishnah just teaches the same rule with regard to olot.