Pesachim Chapter 9, Mishnah 10. This mishnah is somewhat complex. It deals with two groups whose pesach offerings became mixed up with one another, and how to ensure that each person ends up registered on the correct pesach offering.
A halachic introduction - a pesach offering cannot be left without owners:
Before we approach the mishnah, we need to be aware of the rule that a pesach offering cannot be left with no ownership at all. Once the offerings became mixed up, the groups cannot simply say that each of them withdraws from its own pesach offering and chooses a different one instead. If that were possible, there would be no need for our mishnah at all, as we will see shortly.
There are two opinions on this matter:
The first opinion: It is enough that some owners remain registered on the offering, and they do not have to be the original owners.
The second opinion: The original owners themselves must remain registered on the offering.
Our mishnah arranges matters in such a way that one of the original owners will always remain on each and every offering.
The words of the mishnah:
"Shtei chaburos shenisarvu pischeihen" - two groups whose pesach offerings became mixed up.
"Eilu moshchin lahen echad v'eilu moshchin lahen echad" - each group takes one of the two offerings for itself, even though they do not know which of them was originally theirs.
"Echad me'eilu ba lo eitzel eilu v'echad me'eilu ba lo eitzel eilu" - one member of the first group moves over to the second, and one member of the second group moves over to the first.
The formula that the groups recite:
"V'kach hem omrim" - each group, which now holds one of the offerings, turns to the member who joined them and says:
"Im shelanu hu haPesach hazeh" - if this is the pesach offering that was originally ours, "yadcha moshecheses es shelcha v'nimneh al shelanu" - you withdraw from the offering on the other side and register with us on this one.
"V'im shelcha hu haPesach hazeh" - and if this is your pesach offering, meaning that we are standing on the offering that is not ours while you are on the correct one, "yadeinu moshecheses es shelanu v'nimninu al shelcha" - we withdraw from the offering on the other side and register on yours.
In this way, however it turns out, each of the offerings has one of its original owners registered on it.
Extending this to several groups:
"V'chein chameish chaburos shel chamishah chamishah, v'shel asarah asarah" - the same law applies not only to two groups. With five groups of five people each, every group has a person who can move over to each of the other groups, and the same is true of groups of ten. "Moshchin lahen echad mikol chaburah v'chaburah, v'chein hem omrim" - each group takes one offering for itself, and they recite the very same formula. The condition for this is that each group have enough members to send one to each of the other groups.
And if that is not possible, the only way around it is to register additional people on the offering who do not belong to any of the groups, as will be discussed in the next mishnah. However, according to the opinion that specifically requires one of the original owners, this solution is problematic, since a person from the general public was never registered on this pesach offering to begin with.