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Why couldn't Korban Pesach (פסח) be missed, requiring Torah (תורה) to create Pesach Sheni for those unable to bring it? The korban represents our fundamental commitment to bris with Hashem (ה׳) and separation from avodah zarah. It transforms us into a tzibbur united by covenant, making participation essential for membership in Am Yisrael.
Korban Pesach (פסח) represents our committing ourselves to bris with Hashem (ה׳). It showed our separation from a"z - "Mishchu u'k'chu" - along lines of Luchos - which were acceptance of entire Torah (תורה). Thus, the KP couldn't be missed - and Torah allowed make-up of Pesach Sheni. We're also careful to convey its message to all of Israel: we communicate it to each of arba banim (of Haggadah) in manner he'll best relate to. The KP is a korban tsibbur in that it makes us into a tsibbur.
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May a man who has already fulfilled his shofar obligation blow shofar for a woman on Rosh Hashanah? The Baal Itur forbids it while Rav Yehuda permits it. The shiur develops that the core dispute is not about violating shvus for another's benefit, but whether a man's blowing creates a "tekiah shel mitzvah" that enables the woman to be yotzei.
Why does the Gemara connect Purim to Tisha B'Av using "just like"—here you *decrease* joy, here you *increase* it? The shiur develops that Purim fixes the sin of Achashverosh's feast, where Klal Yisrael's desire to socialize with goyim revealed a physical disconnection from each other. Mishloach Manot and matanos la'evyonim create the physical unity—eating and drinking together—that is the tikkun for sinas chinam and the foundation for Kabbalas HaTorah.
