Aside from the korban Pesah, generally speaking during the time of the Temple the people also sacrificed a korban hagiga – a special sacrifice in honor of the holiday – which was also eaten during the seder. The Mishna (69b) teaches that the korban hagiga was brought only if it was a weekday, the people were ritually pure and there would not be enough meat from the korban Pesah to satisfy everyone in the group. Were erev Pesah to fall out on Shabbat, or if the people were tameh – ritually defiled, and the korban Pesah was being brought only because of the principle of tuma hutra be-tzibur (see above, daf 67) – or if not many people had joined together to participate in this korban Pesah, so there would be enough meat for all, then the korban hagiga would not be brought.
Rav Ashi points out an obvious conclusion based on this Mishna. Clearly the korban hagiga is not obligatory on the 14th of Nisan (erev Pesah), since if it had to be brought then it would “push aside” Shabbat and the rule of tuma hutrah be-tzibur would be applied. Similarly, the number of people partaking of the korban Pesah would be irrelevant.
The Gemara asks: If there is no obligation to bring this offering, what is the reason that it nevertheless comes when each person’s portion of the Paschal lamb is small? The Gemara explains that the reason is as it was taught in a baraita: The Festival peace-offering that comes with the Paschal lamb is eaten first; the reason for this is so that the Paschal lamb will be eaten when one is already satiated. The Paschal lamb should not be eaten in a needy manner, but rather in joy and when one is already filled to satisfaction.
The Jerusalem Talmud explains that the korban Pesah is eaten al ha-sova (on a full stomach) in order to ensure that the people are not hungry and will not come to break the bones of the korban Pesah in order to eat the marrow, since breaking a bone from the korban Pesah is forbidden by the Torah (see Shmot 12:46). The Mordechai explains that really all Temple sacrifices are supposed to be eaten al ha-sova, since while eating them we are supposed to perceive ourselves as royalty, and our meal should not be one of ravenous hunger but one that gives a sense of plenty.