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 Summary 

 Mishna: 

The mitzvos of lulav and aravah may be fulfilled for six or seven days.

Hallel and simcha [eating shlamim from the table of Hakodosh Boruch Hu] are obligatory on all eight days [Tosfos: While it is not always possible to eat shlamim meat on all eight days, it is possible to fulfill the mitzvah of simcha with fine wine or by another method Click here to listen to the shiur with Tosfos]. https://drive.google.com/file/d/1myTrYrVn00U6X7NM4Gpjs-q468SVhyiV/view?usp=sharing_

The mitzvos of sukkah and nisuch hamayim are obligatory for seven days. 

The flute [the music at the simchas beis hashoeiva] may be fulfilled for five or six days [since it is not done on Yom Tov or on Shabbos]. 

 Lulav is seven days if the first day is Shabbos, on other days the lulav was not taken on Shabbos.

 Aravah is seven days if the last day is Shabbos, on other days the aravah was not taken on Shabbos.

If the first day of Yom Tov was on Shabbos, everyone carried their lulav to Har Habayis before Shabbos and the shamashim laid them all out on a platform. The older people placed theirs in a room. The people were taught to say “Anybody who will get my lulav, let him keep it as a gift from me”. The next day they would come early and the shamashim would throw the lulavim towards them and they would scramble to grab a lulav [this was a good procedure for a long time] but sometimes they hit each other [in the end], when Beis Din saw that this causes a sakanah , they instituted for everyone to fulfill the mitzvah in their own homes.