Mishnah 4 in chapter 1 lists a series of technical requirements that must be met for a court to be fit to bring a par helem davar. Even when the ruling itself would otherwise create the obligation, a flaw in any one of these details is enough to prevent the offering from being brought.
The first detail - a judge who said "you are mistaken":
If the Sanhedrin issued its ruling, and one of the seventy-one judges knew it was an error and said explicitly, "you are mistaken" - the ruling still stands, since the rule is that we follow the majority, and there is nothing invalid about the existence of a dissenting opinion; that is the natural course of a discussion in a court, and certainly in the Sanhedrin. However, the verse requires that this be "the entire congregation," meaning here the members of the Sanhedrin who serve as the agents of the public in issuing the ruling. Since not all of them agreed to the ruling, for a technical reason there is no possibility of bringing a par helem davar, even when they erred in a ruling concerning a matter whose intentional violation carries kareis and the like.
The second detail - a ruling issued without consulting the mufla:
If the decision was not based on consulting the opinion of the mufla of the court - the leading personality, the greatest sage of the generation, the outstanding Torah scholar of the Sanhedrin - the matter is not considered a complete inadvertence, but comes close to being deliberate. This is not a mere mistake and not an accident, but negligence: how could they possibly fail to consult the greatest member of the court?
The case under discussion is as follows: certainly the Sanhedrin does not judge with only seventy members, and all seventy-one are required. But it is possible that the greatest sage of the generation had to leave on an important mission, and another judge was appointed in his place, so that they had a full quorum of seventy-one. Even so, since the mufla was not a partner to the decision, this is not a responsible decision but something close to negligence, and therefore - even if all the other conditions were met - it is not considered a true inadvertence, and for a technical reason no par helem davar is brought.
In both of these cases, and likewise in the cases that follow, the result is the same: if the majority of the Jewish people (as will be defined later) indeed followed the ruling of the Sanhedrin and transgressed a sin whose intentional violation carries kareis, each and every one of them brings his own sin offering. Since no par helem davar is brought on account of those technical disqualifications in the Sanhedrin, the obligation reverts to every single individual to bring his own private sin offering.
The third detail - a judge who is disqualified from judging capital cases:
The Mishnah continues: "O she'hayah echad mehen ger o mamzer o nasin o zaken she'lo ra'ah lo banim" - that is, if one of the members of the Sanhedrin was someone who is not fit to judge capital cases.
This requires explanation, and it is not as simple as it appears at first glance. The rulings of the Sanhedrin that lead to a par helem davar are generally in matters of what is permitted and forbidden: a question in the laws of Shabbos, in the laws of kashrus, or in the laws of niddah. Seemingly there is no need for a judge who has children to sit and rule on questions of this sort, and even an individual or a court of three can rule on them. But a gezeirah shavah teaches that if the members of the court are not fit to judge capital cases, their ruling does not create an obligation of a par helem davar. Therefore, if someone who is not permitted to judge capital cases sat among them - even if his sitting in the Sanhedrin was legitimate - for a technical reason no par helem davar is brought.
A distinction must be drawn between the first three disqualifications and the fourth:
A convert, a mamzer and a nasin: Their disqualification is derived from the verse said to Moshe Rabbeinu at the establishment of the Sanhedrin - "and they shall stand there with you": the seventy elders stand with you, meaning that they are of distinguished lineage like you. The convert - whose lineage is not the lineage of a born Jew, whether he himself is a convert or his parents are converts; the mamzer - born of a forbidden union punishable by kareis, excluding niddah; and the nasin - from the Givonim, who were a Canaanite tribe and deceived Yehoshua regarding their identity until he made a covenant with them. None of these has any place in the Sanhedrin at all, and if they were present in it, it is not a legitimate Sanhedrin on whose account a par helem davar is brought.
An elder who has never had children: "Elder" here means a judge, and not necessarily an old man, and the reference is to someone who never had children. This disqualification has an entirely separate derivation: one who never had children does not know a parent's mercy and compassion for his child, and therefore he may not sit on a court that judges capital cases - the small Sanhedrin of twenty-three. Although he is permitted to sit in the Sanhedrin and rule on matters of what is permitted and forbidden, for a technical reason, if he was among those seventy-one, the Sanhedrin is not fit to bring a par helem davar.
The Source of the Law - a Gezeirah Shavah:
The Mishnah brings the derivation that links the requirements of a judge who rules in capital cases, where a man may be put to death, with the requirements of the members of the Sanhedrin who bring the bull for a mistaken ruling: "shene'emar kan edah, vene'emar lehalan edah" - the word "edah" (congregation) is stated here, and the word "edah" is stated there. In the passage of the bull for a mistaken ruling it says, "And if the whole congregation of Israel shall err," and in capital cases it says, "and the congregation shall judge," and it says, "and the congregation shall save" - that same well known verse dealing with the procedures of a court that rules in capital cases.
Since the word "edah" appears in both places, they are derived one from the other through a gezeirah shavah: "mah edah ha'amur lehalan ad shekulam re'uyin lehora'ah, af edah ha'amur kan ad shekulam re'uyin lehora'ah" - just as the "congregation" mentioned there requires that all of them be fit to issue rulings, so too the "congregation" mentioned here requires that all of them be fit to issue rulings. Just as there, in capital cases, every member of the small Sanhedrin of twenty three must be qualified - including that they have children - so too with the congregation mentioned here, the Sanhedrin of seventy one: all of them must be fit to judge capital cases, they must have children, and certainly none of them may be a convert, a mamzer or a nasin.
The Court Ruled and Erred - the Three Situations:
"Horu beis din veshagegu ve'asu kol hakahal shogegin": the court issued its ruling and made an honest mistake, the people acted according to its ruling in good faith, relying upon it, the majority of the community transgressed, and afterwards the court retracted - "mevi'in par" - they bring a bull. This is the classic case of the bull for a mistaken ruling.
"Horu mezidin ve'asu shogegin": the Sanhedrin deliberately ruled that a certain thing was permitted while knowing that it was forbidden, for whatever reason - for example, that the cheilev dealers in the country bribed the judges to rule that cheilev is permitted - and the people followed the ruling and believed in good faith that the thing was permitted. In this case, "mevi'in kisbah o se'irah" - they bring a female lamb or a female goat: no bull for a mistaken ruling is brought, but rather each individual who unwittingly transgressed a sin whose deliberate violation carries kareis, such as eating cheilev, brings an individual sin offering - a kisbah (a female lamb) or a se'irah (a female goat), for an individual sin offering comes from a lamb or a goat, a female.
"Horu shogegin ve'asu mezidin": the Sanhedrin erred in good faith, but the people followed it while knowing full well that it had erred - "harei eilu peturin" - these are exempt.
An explanation of the third situation: one must distinguish between a cause and an excuse. Here it was not the ruling of the Sanhedrin that brought the people to transgress; it merely served them as an excuse. For example, the Sanhedrin's Torah scroll was invalid and nobody knew of it, and on its basis they permitted something forbidden. The Sanhedrin erred innocently; but if the people said, "We know that the Torah scroll in their hands is invalid, but let us not reveal it to them - after all, they have permitted it, so let us go ahead and do it" - it turns out that they transgressed not because of the Sanhedrin's ruling, but because of their own evil decisions and their corrupt character.
Therefore nobody brings an offering. The Sanhedrin does not bring the bull for a mistaken ruling, since the people did not rely upon it but upon their own evil decisions and their own desires, and its ruling was not the cause of the sin but merely an excuse, and it requires no atonement, for the sin does not rest upon it. And the people do not bring a sin offering, for a sin offering comes only for an act done unwittingly, and they acted deliberately, and a deliberate sin is not atoned for by a sin offering.