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The name and its reason:
The Shabbos before Pesach is called Shabbos Hagadol, as on that Shabbos a great miracle occurred. On the 10th of Nissan was the day that Bnei Yisrael took a lamb for the Pesach offering, as is stated in the verse “On the tenth of this month, they are each to take a lamb on behalf of their families.” Now, that day was Shabbos. When the Jewish people took their lambs that Shabbos, the first-born gentiles gathered by Bnei Yisrael and asked for an explanation for their actions. Bnei Yisrael told them that the lamb is a Pesach sacrifice to G-d, as He will kill the Egyptian first-borns. [Upon hearing this] the firstborns proceeded to go to their fathers and to Pharaoh and request from them to release the Jewish people. When they were answered negatively, the first-borns resorted to making war with Pharaoh, and they killed many of the Egyptians. This is the meaning of the verse “And Egypt was stricken by their first born”. This miracle was established to be remembered for all generations on the Shabbos [before Pesach], and they called it “The Great Shabbos”. [Shabbos Hagadol was thus the beginning of the redemption and the miracles.]
The Haftorah when Shabbos Hagadol coincides with on Erev Pesach:
The custom in these provinces is that when Shabbos Hagadol falls on Erev Pesach the Haftorah of “Veareiva..” is read from Malachi.
The Shabbos Hagadol speech: Drasha:
It is customary for the community Rav to give a speech dealing with the laws of the Pesach on the Shabbos preceding the festival.
The content of the speech: The main point of this speech is to expound and teach the people the ways of G-d, and teach them what must be done [on the holiday], unlike the custom today [that the speech surrounds other topics].
When is the speech given if Shabbos Hagadol falls on Erev Pesach? When Erev Pesach falls on the Shabbos prior to Pesach, the speech is given on the previous Shabbos.
It is customary for the community Rabbi to give a speech dealing with the laws of the Holiday on the Shabbos preceding Pesach, [unless that Shabbos is Erev Pesach, in which case it is given on the preceding Shabbos]
Saying Avadim Hayinu:
The custom in these provinces is to recite [after] Mincha of Shabbos Hagadol, “Avadim Hayinu” [until Lichaper Al Kol Avonoseinu]. Even when Shabbos Hagadol falls on Erev Pesach, one says Avadim Hayinu.
Vayehi Noam by Maariv of Motzei Shabbos:
Whenever Pesach, or any Yom Tov, falls on a weekday, the paragraph of Vayehi Noam, which is normally said after Shabbos, is omitted.
If Pesach falls on Shabbos, is Vayehi Noam recited on the Motzei Shabbos prior to it?
 Admur 430:1-3
 Admur 430:1; Michaber 430:1; Tosafos Shabbos 87b
Other reasons: Another reason for the name “Shabbos Hagadol” is because the Drasha on this Shabbos takes a long time, and therefore the congregation ends up delaying the leave of Shabbos, and it seems like a longer Shabbos. It is thus called the great Shabbos. [Shivleiy Haleket, brought in Tzemach Tzedek]
Why the miracle commemorated on Shabbos and not on the weekday of its occurrence: The reason why this miracle was not established on the 10th of the month, whether it falls on Shabbos and whether it falls on a weekday, as is the case with all other commemorated events, is because Miriam died on the 10th of Nissan, and this date was instituted as a fast day, in years that the 10th of Nissan occurs on a weekday, as brought in 580:2. [Admur ibid; M”A 430:1; Thus, due to the preoccupation of this date with the passing of Miriam, the commemoration of the Shabbos Hagadol miracle was set for the weekday of its occurrence, which was the Shabbos before Pesach.]
 Shemos 12:3
 How we know that the 10th of Nissan originally fell on Shabbos? The Jewish people left Egypt on the 15th of Nissan, which was on a Thursday, as written in 494:1 [and Shabbos 87b], thus the 10th of Nissan was on the previous Shabbos. [Admur ibid]
 Tehillim 136:10
 See Admur 430:2; Likkutei Sichos 37 p. 10
 Admur 430:3; Beir Heiytiv 430:1; Or Zarua 2:393; See Piskeiy Teshuvos 430:1
 The reason: As this Parsha discusses the bringing of the tithes to the storehouses, which is similar to what occurs on Erev Pesach, being that on Erev Pesach is the time of Biur. As on every Erev Pesach of the 4th and 7th year of the Shemitah it is an obligation upon every person to remove from his house all of the Maaseros of grains that he tithed in the past three years of Shemitah and that have remained in his house. On this Erev Pesach, he is obligated to remove it from his house and bring it to the house of the Levites. (Now, although there are those [Rambam Maaser Sheiyni 11:7; Sifri Devarim 14:28] who say that the time for this obligation is not on Erev Pesach but rather on Erev Shevi’i Shel Pesach, and so is the main Halachic ruling as brought in Y.D. 331:144, nevertheless, the widespread custom is like the former opinion.) [Admur ibid]
 Admur 429:2; M”A 429:1; Bach 429; Rashi Bechoros 60a
The background of this custom from 429:2:
[It has already been stated above that the Sages in the times of the Temple instituted that all are to begin studying the laws of the festival thirty days prior to the festival. After the Temples destruction, only the students continued this custom, and were taught the laws thirty days prior to the festival. However] for the community, the laws of the upcoming festival would be expounded on the Shabbos prior to the festival, as on this Shabbos all the Jews from the villages would gather to hear the laws of the festival from the Rabbi. For this reason, in the later generations, the custom became [even though today this concept of ingathering no longer applies] that the Rabbi expound on the laws of Pesach on the Shabbos before it, and the laws of the Chag (Sukkos) on Shabbos Shuvah. [Admur ibid; Accordingly, the teaching of the laws thirty days prior did become nullified with regards to the townspeople which are not students. Seemingly, this shows that today there is no true obligation to teach thirty days prior, but rather just a custom to teach the laws beforehand.]
 Admur ibid; M”A 429:20 and 290; See also Admur 290:3 and Hilchos Talmud Torah 4:4
 Regarding the meaning of the phrase “to expound and teach the people the ways of G-d and teach them what must be done [on the holiday]” In the source of Admur in the Magen Avraham there is no “and” between “ways of G-d” and “teach..” Thus, one can interpret that teaching the laws is the meaning behind teaching them the ways of G-d. However, in Admur, since he adds an “and” between these two phrases, it must be that they are two different matters. The meaning behind “ways of G-d” refer to discussing general teachings in service of G-d. It is for this reason that Admur does not mention that the laws of Yom Kippur have to be also discussed on Shabbos Shuvah, as the main laws of Yom Kippur [which is Teshuvah] is already included in the teachings of the path of serving G-d, that is mentioned in the first phrase, and thus there was no need to mention it, just like we do not say the laws before Shavuos being that they are already included. [Likkutei Sichos vol. 3 Shabbos Hagadol]
 Admur ibid; Maharil; Chok Yaakov 429:3; Elya Raba 430:2
 Admur 430:2; Rama 430:1; Minhagim Tirana; Rebbe in Haggadah Shel Pesach
Other Opinions: Some Poskim rule that one does not say Avadim Hayinu on Shabbos Hagadol. [Gra; Omitted from the Siddur of the Arizal]
Chabad custom: The Shaar Hakolel [48:1] writes that from the fact that Admur omitted this in his Siddur, it is evident that in his Siddur he holds that one does not say it on Shabbos Hagadol. Nonetheless, the Rebbe writes in his commentary on the Haggadah that the custom is to say it, unlike the ruling of the Shaar Hakolel.
 Rebbe in his commentary on the Haggadah
Other Opinions: From Admur in his Shulchan Aruch ibid, it is implied that Avadim Hayinu is to be said after the Chazzans repetition of the Amidah, as Admur states “In place of saying Barchi Nafshi (which is not said according to our Nussach) one says Avadim Hayinu” and Barchi Nafshi is said before Aleinu. [see Siddur Yavetz]
 Rebbe in his commentary on the Haggadah.
Opinion of Admur: Admur in his Shulchan Aruch does not write until where in the Haggadah one should read.
 The reason: As Shabbos Hagadol was the beginning of the redemption and the miracles. [Admur ibid; Levush 430]
 Admur ibid; Darkei Moshe 430; Sefer Haminhagim p. 33
 Admur 295:3
 Implication of Admur 295:3; Peri Megadim 295 M”Z 2; Poskim brought in Shaareiy Teshuvah 295:2; M”B 295:3
The Chabad Custom: Rav Yaakov Landau writes in his diary that the Rebbe Rashab would recite Vayehi Noam in such a case. Likewise, Rav Zalman Shimon Dworkin states that the custom in Lubavitch was to say it. [Shevach Hamoadim p. 184] However, in the Luach Kolel Chabad, they write that it should not be said. See Nitei Gavriel Pesach 1, and Otzer Minhagei Chabad Nissan.
The Rebbe’s custom: Some record that the Rebbe did recite Vayehi Noam in such an instance. [Kitzur Halachos 293 footnote 9] However, Rabbi Leibal Groner replied to me that in 1988, when the Rebbe Davened for the Amud, he did not say it, and rather said Kaddish Tiskabel.
 Sheialas Yaavetz 19 brought in Shaareiy Teshuvah 295:2; Kaf Hachaim 295:7; Minchas Shabbos 96:7; Luach Tukichinsky writes that so is the custom in Eretz Yisrael;
 The reason: It should not be said when Pesach falls on Shabbos, as Erev Pesach is considered a holiday, being that it’s forbidden to do work from Chatzos. [Poskim ibid]