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The Gemmorah (Yevomos 62b) brings a statement by R'Yehoshua:

...דתניא: רבי יהושע אומר נשא אדם אשה בילדותו ישא אשה בזקנותו היו לו בנים בילדותו יהיו לו בנים בזקנותו שנא' (קהלת יא, ו): "בבקר זרע את זרעך ולערב אל תנח ידך", כי אינך יודע אי זה יכשר הזה או זה ואם שניהם כאחד טובים

...Rabbi Yehoshua says: ... If he had children in his youth, he should have more children in his old age, as it is stated: “In the morning sow your seed, and in the evening do not withhold your hand; for you do not know which shall prosper, whether this or that, or whether they both alike shall be good” (Ecclesiastes 11:6). This verse indicates that a man should continue having children even after he has fulfilled the Mitzvah to be fruitful and multiply.

Do we have sources that discuss how many kids did the Rabbis (in the times of the Mishna and Gemmorah) had on average and their reasons for that?

This question is not about specific Rabbis, as we know about R' Akivah, R' Meir or R' Papa, but general trends and statistics.

  • @JoelK דין גרמא דעשירי ביר? I thought of that, and it seems that he was an exception, do we have another example of that many kids? (The ten names that we mention in Syium Mashecht aren't all Rav Papa sons) – Alaychem Aug 26 at 6:57
  • @Alaychem This was the first example that came to mind. I have no idea if it's representative... – Joel K Aug 26 at 7:01
  • Also tosafos nidah 8a veamar @Joelk – Dr. Shmuel Aug 28 at 3:54
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Not a concrete source, but there is well known information about Rav Papa’s children:

חַנִינָא בַּר פַּפָּא, רַמִי בַּר פַּפָּא, נַחְמָן בַּר פַּפָּא, אַחַאי בַּר פַּפָּא, אַבָּא בַּר פַּפָּא, רַפֽרַם בַּר פַּפָּא, רָכִיש בַּר פַּפָּא, סוֹרְחָב בַּר פַּפָּא, אַדָא בַּר פַּפָּא, דָרוֹ בַּר פַּפָּא

This was not even all of his children per se (See here). Also, there is to differentiate between successful children and those who died in infancy.

  • Like Avos Rebbi nason 14.6 for example – Dr. Shmuel Aug 28 at 5:34
  • Rabbi Yehuda Herzl Henkin points out that no women back then could have (or survive) ten pregnancies; ten sons would have meant with at least two different wives -- probably not polygamy, just remarrying after a spouse died. – Shalom Aug 28 at 7:36
  • Sefer haeshkol entertains that as well – Dr. Shmuel Aug 28 at 9:53

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