Who was Nasi and who was Av Bais Din?

Question

The third zug was Shimon ben Shetach and Yehuda ben Tabai. Wondering who from these two sages was the Nasi and who was the Av Bais Din?

Answer 1

It's actually a dispute in the Gemara:

מַתְנִי׳ יוֹסֵי בֶּן יוֹעֶזֶר אוֹמֵר שֶׁלֹּא לִסְמוֹךְ, יוֹסֵף בֶּן יוֹחָנָן אוֹמֵר לִסְמוֹךְ. יְהוֹשֻׁעַ בֶּן פְּרַחְיָה אוֹמֵר שֶׁלֹּא לִסְמוֹךְ, נִיתַּאי הַאַרְבֵּלִי אוֹמֵר לִסְמוֹךְ. יְהוּדָה בֶּן טָבַאי אוֹמֵר שֶׁלֹּא לִסְמוֹךְ, שִׁמְעוֹן בֶּן שָׁטַח אוֹמֵר לִסְמוֹךְ. שְׁמַעְיָה אוֹמֵר לִסְמוֹךְ, אַבְטַלְיוֹן אוֹמֵר שֶׁלֹּא לִסְמוֹךְ. הִלֵּל וּמְנַחֵם לֹא נֶחְלְקוּ. יָצָא מְנַחֵם, נִכְנַס שַׁמַּאי. שַׁמַּאי אוֹמֵר שֶׁלֹּא לִסְמוֹךְ, הִלֵּל אוֹמֵר לִסְמוֹךְ.
MISHNA: Yosei ben Yo’ezer says not to place one’s hands on offerings before slaughtering them on a Festival because this is considered performing labor with an animal on a Festival. His colleague, Yosef ben Yoḥanan, says to place them; Yehoshua ben Peraḥya says not to place them; Nitai HaArbeli says to place them; Yehuda ben Tabbai says not to place them; Shimon ben Shataḥ says to place them; Shemaya says to place them; Avtalyon says not to place them. Hillel and Menaḥem did not disagree with regard to this issue. Menaḥem departed from his post, and Shammai entered in his stead. Shammai says not to place them; Hillel says to place them.
הָרִאשׁוֹנִים הָיוּ נְשִׂיאִים, וּשְׁנִיִּים לָהֶם אֲבוֹת בֵּית דִּין.
The first members of each pair served as Nasi, and their counterparts served as deputy Nasi.
גְּמָ׳ תָּנוּ רַבָּנַן: שְׁלֹשָׁה מִזּוּגוֹת הָרִאשׁוֹנִים שֶׁאָמְרוּ שֶׁלֹּא לִסְמוֹךְ, וּשְׁנַיִם מִזּוּגוֹת הָאַחֲרוֹנִים שֶׁאָמְרוּ לִסְמוֹךְ — (הָרִאשׁוֹנִים) הָיוּ נְשִׂיאִים, וּשְׁנִיִּים לָהֶם אֲבוֹת בֵּית דִּין, דִּבְרֵי רַבִּי מֵאִיר. וַחֲכָמִים אוֹמְרִים: יְהוּדָה בֶּן טָבַאי — אַב בֵּית דִּין, וְשִׁמְעוֹן בֶּן שָׁטַח — נָשִׂיא.
GEMARA: The Sages taught: Three of the first pairs who say not to place hands and two of the last pairs who say to place hands served as Nasi, and their counterparts served as deputy Nasi; this is the statement of Rabbi Meir. And the Rabbis say the opposite: Yehuda ben Tabbai was deputy Nasi and Shimon ben Shataḥ was the Nasi.

So the Mishnah follows the opinion of R' Meir that Yehuda ben Tabai was Nasi (as Mishnayos generally follow the opinion of R' Meir), but the Sages held that Shimon ben Shetach was the Nasi.