A minyan is a gathering of ten male Jews of sound mind and responsibility to join together for matters of sanctity. A minor, who is not yet of full sound mind and competence, does not count as part of the minyan. When he reaches the age at which he is obligated to fulfill the mitzvot (bar mitzvah), he is counted as part of the minyan.
There are some Rishonim who maintain that in extenuating circumstances (b’sha’at hadchak), nine adults may include a minor who is holding a Chumash in his hand as part of their minyan. However, in the opinion of most poskim, even in extenuating circumstances, a minor may not be considered part of the minyan, and that is how we practice. Nevertheless, in a situation in which the minyan will be canceled completely unless he is counted, possibly causing some of the members to distance themselves from Judaism, he can be counted as part of the minyan.
Similarly, a shoteh (deranged person), who is not of sound mind – for example, one who takes off his clothes and ruins them – cannot be counted as part of a minyan. Regarding one who temporarily loses his sanity but afterwards returns to his senses, when he loses his sanity – he cannot be included as part of the minyan and when he regains his sanity – he may be counted (Chagigah 3b; Bei’ur Halachah 55:8). Likewise, one who is drunk like Lot, and does not know what is being done to him, may not be included in a minyan. L’chatchilah, it is preferable not to count a drunken person who is incapable of speaking before the King as part of the minyan (Kaf HaChaim 55:14; and see further in this book 5:11).
Since a person who is deaf-mute does not have a way of communicating with the world, the Chachamim say that he is considered a shoteh and he is exempt from fulfilling the mitzvot. Therefore, he may not be counted as part of the minyan (Chagigah 2b; Shulchan Aruch 55:8). The poskim disagree as to whether or not the status of a deaf-mute person changes, if he was taught to communicate with his surroundings, either by sign language or by reading and writing. Since joining a minyan is a rabbinic mitzvah, the halachah follows those poskim who are lenient, and therefore he may be counted in a minyan.
. The Amora’im are divided as to whether or not a minor may be counted as part of the minyan. The Rishonim dispute this as well, as brought in the Beit Yosef, Orach Chaim 55:4. The overwhelming majority of the Rishonim, including the Rambam, Ramban, Rashba, and Rosh, write that he may not be counted. However, Rabbeinu Tam, the Razah, and others write that we may be lenient in extenuating circumstances. Yabia Omer, part 4, 9, summarizes the issue and concludes that it is prohibited for a minor to be included even in extenuating circumstances. Nonetheless, the Maharsham, responsa 3:164, and Igrot Moshe, Orach Chaim, part 2, 18, permit it in extenuating circumstances because the law of prayer in a minyan is rabbinic and in certain situations we can rely on a minority of poskim.
The definition of an adult is one who has reached thirteen years of age and two hairs have grown in. Yet, practically we are not strict about this because praying in a minyan is a rabbinic obligation and we rely on the presumption that he has two hairs. Even if in actuality he does not, perhaps he did and they shed. See Mishnah Berurah 55, passages 31 and 40.
. According to the Tzemach Tzedek 73 and Divrei Chaim, part 2, Even HaEzer, 72, even if the deaf-mute is a very wise person, he is exempt from the mitzvot. According to Maharsham, part 2, 140, and Nachalat Tzvi, he is considered a competent person for all intents and purposes. A few Acharonim write that because minyan is a rabbinic ruling, we may rely on those who are lenient and count him, as it is written in the responsa of Nachalat Binyamin 31 and Yechaveh Da’at 2:6. In a minyan of this kind, it is better not to have the chazan repeat the Amidah, for some poskim maintain that if there are not nine people answering, the berachot are close to being in vain. See further on this topic in Nishmat Avraham, Orach Chaim 1, 55:2, and Encyclopedia of Jewish Medical Ethics, entry “deafness.” Additionally, if the person who is deaf-mute was taught to make sounds that are comprehensible to most people, according to all opinions, he may be counted in a minyan (Halichot Shlomo 22:26).