As we learned in the first Mishna (daf 2a), if a sacrifice were brought she-lo li-shmah – with the wrong intention in mind, e.g. the animal had been set aside for one type of sacrifice but was slaughtered for a different sacrifice – it remains a valid sacrifice, although it does not count and the owner will need to bring another sacrifice to fulfill his obligation. The only exceptions are the korban Pesaḥ and a korban ḥattat – a sin-offering – which must be brought with the proper intent.
Shimon ben Azzai concludes the discussion by saying “I have a tradition from the 72 elders, on the day that Rabbi Elazar ben Azarya was appointed head of the yeshiva, that all edible sacrifices that were brought without proper intent are valid, although they will not count towards the obligation of their owners, with the exception of the korban Pesaḥ and korban ḥattat, which must be brought with the proper intent.” The Mishna points out that Shimon ben Azzai’s teaching places a korban olah – a burnt offering – in the same category as the korban Pesaḥ and the ḥattat, inasmuch as it is not an edible sacrifice (it is entirely burned on the altar, and none of it is eaten). Nevertheless, the Sages of the Mishna reject his tradition.
When Shimon ben Azzai refers to the day that Rabbi Elazar ben Azarya was appointed head of the yeshiva, he is talking about a famous situation described in Gemara Berakhot (27b-28a). The head of the academy at that time was Rabban Gamliel, whose behavior towards Rabbi Yehoshua was perceived by the students as lacking in respect. In response to this, the students deposed Rabban Gamliel and elected Rabbi Elazar ben Azarya to head the yeshiva in his place. The Gemara there relates that Rabbi Elazar ben Azarya opened the gates of the academy to all, removing the restrictions on acceptance instituted by Rabban Gamliel, and that all of the Sages engaged in a review of questions that had been left undecided at that time, reaching conclusions with regard to all of them.
The Ri”d suggests that the 72 elders mentioned by Shimon ben Azzai includes the 71 members of the Sanhedrin in Yavne, as well as the muflah shel beit din – the elder who participated in the discussion without being a formal member of the group (see Massekhet Horayot daf 4). Some suggest that the additional individual was Rabban Gamliel, who, according to the Gemara in Berakhot, did not take umbrage at the coup and participated fully in those discussions.