Quoting Commentary for Pirkei Avot 5:21

הוּא הָיָה אוֹמֵר, בֶּן חָמֵשׁ שָׁנִים לַמִּקְרָא, בֶּן עֶשֶׂר לַמִּשְׁנָה, בֶּן שְׁלשׁ עֶשְׂרֵה לַמִּצְוֹת, בֶּן חֲמֵשׁ עֶשְׂרֵה לַתַּלְמוּד, בֶּן שְׁמֹנֶה עֶשְׂרֵה לַחֻפָּה, בֶּן עֶשְׂרִים לִרְדֹּף, בֶּן שְׁלשִׁים לַכֹּחַ, בֶּן אַרְבָּעִים לַבִּינָה, בֶּן חֲמִשִּׁים לָעֵצָה, בֶּן שִׁשִּׁים לַזִּקְנָה, בֶּן שִׁבְעִים לַשֵּׂיבָה, בֶּן שְׁמֹנִים לַגְּבוּרָה, בֶּן תִּשְׁעִים לָשׁוּחַ, בֶּן מֵאָה כְּאִלּוּ מֵת וְעָבַר וּבָטֵל מִן הָעוֹלָם:

He was wont to say: Five years old for (learning) Scripture. [This is derived from arlah (forbidden fruit), viz. (Leviticus 19:23): "Three years shall it be forbidden to you… (24): And in the fourth year all of the fruit shall be holy for praise to the L rd." (At this stage) his father teaches him the form of the letters and the vowel signs. (25): "And in the fifth year you may eat its fruit, to increase for you its produce." Thenceforward, stuff him like an ox!], ten years old for Mishnah. [He learns Scripture for five years and Mishnah for five years, viz. (Chullin 24a): The master said: Any student who has not made progress in his Mishnah after five years, will no longer do so, as it is written (Numbers 8:24): "This is what applies to the Levites. From the age of twenty-five and up shall he come to enter into service to work in the tent of meeting." He learns the halachoth of the service for five years, and, at thirty, he works.], thirteen years old for mitzvoth, [it being written (Numbers 5:6): "A man or a woman if they do of all the sins of man" and, in respect to (the episode of ) Shchem, (Genesis 34:25): "And two sons of Jacob, Shimon and Levi, the brothers of Dinah, took, each man his sword, etc." Levi at that time was thirteen years old and he is called "a man"], eighteen for the (marriage) canopy. ["Adam" ("man") is written nineteen times in the parshah of Bershith — from (1:26): "And G-d said: "Let us make a man" until (2:22): "And the L rd G-d fashioned the side that He had taken from the man into a woman" — one time for (the plain meaning) itself, leaving eighteen for expounding ("derashah").], twenty years old for pursuing [a livelihood. After he has learned Scripture, Mishnah, and Gemara, and taken a wife and begotten children, he must pursue a livelihood. Or: "twenty years old for pursuing" him from Heaven and punishing him for his acts, the Heavenly tribunal not punishing one who is less than twenty years old.], thirty for strength, [the Levites setting up the tabernacle and taking it apart and loading the wagons and carrying on their shoulder from the age of thirty on], forty years old for understanding, [for after forty years that Israel was in the desert Moses said to them (Deuteronomy 29:3): "and the L rd did not give you a heart to know and eyes to see and ears to hear until this day."], fifty years old for counsel, [it being stated of the Levites (Numbers 8:25-26): "And from the age of fifty he shall return from the service of the work, and he shall serve no more. And he shall serve his brothers." Which service? Counsel.], sixty years old for old age, viz. (Iyyov 5:26): "You will go to the grave in old age (bekelach): — the gematria (numerical equivalent) of "bekalach" is sixty.], seventy years old for hoary old age, [it being written in respect to David (I Chronicles 29:28): "And he died in a good hoary old age, and the days of his life were seventy years."], eighty years old for strength, [it being written (Psalms 90:10): "And if with 'strengths,' eighty years."], ninety years old for bowing (shuach) [He walks bowed and bent over. Some see it as relating to "shuchah amukah," "a deep pit."], one hundred years old — as if he had died and passed on and vanished from the world.

Rashi

נשא את ראש וגו׳ TAKE THE SUM OF [THE SONS OF KOHATH] — Count those that are fit among them for the service of carrying, and these are the men from thirty to fifty years, for a person less than thirty has not yet attained his full strength, — hence they (the Rabbis) said (Avot 5:21) “a man of thirty has reached the age when he enters into full strength”, — as for one who is older than fifty, his strength gradually diminishes from then.
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Or HaChaim

The Torah also wanted to give a hint to the elite amongst the people to influence the hearts of the Israelites so that they would worship G'd willingly. The reason that the offering or sacrifice is called קרבן, i.e. from the root קרב, "to come near," is that it is sin which separates man from his Father in Heaven. We know from Deuteronomy, 4,4 that "you who have cleaved to the Lord are all alive this day." The Israelites instead had become separated and distant from the presence of G'd, the שכינה, due to their support for the spies' report and their attempt to return to Egypt. G'd, on the other hand, is anxious to establish a close bond with Israel and commanded the elite to rebuke the sinner in order to bring his heart closer to the Lord his G'd. At the same time G'd punished those who ignored their duty in this regard. A proof that G'd indeed expects those who are the elite to take a leading role in influencing the hearts of the people, is found in the statement by our sages in Avot 5,21: "He who leads the multitude in righteousness shall have no sin come into his hand," seeing G'd will protect him against committing errors. These then are the messages contained in our verse.
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Ramban

FROM TWENTY YEARS OLD AND UPWARD, ‘KOL YOTZEI TZAVA’ (ALL THAT ARE ABLE TO GO FORTH TO THE HOST) IN ISRAEL. “This tells us that no one under the age of twenty goes forth to the host. ‘Se’u eth rosh’ [literally: ‘take the head’ and generally translated: take ye the sum] of all the congregation of Israel.16Verse 2. This is as one says to an executioner: ‘take that man’s head’.”17The meaning of this text which stems from Bamidbar Rabbah 1:9 will be explained further on by Ramban. The final quote is not found in our texts of Rashi, although Ramban quotes it from his commentary. The reason for its disappearance from all other texts of Rashi may well be its sharpness of expression which on first sight baffles the student. Ramban, however, recognizing that it is a genuine text which stems from Midrash Rabbah proceeds to explain it appropriately. This is Rashi’s language.
Perhaps the reason for this [law that a male under twenty years old was not liable to military service] is because he is not strong enough for war under the age of twenty, as the Rabbis have said:18Aboth 5:21. “Twenty is the age for pursuit.”19The Hebrew is lirdof, which is variously interpreted as: “for seeking a livelihood:” or “for seeking one’s life-pursuit.” Ramban here understands it in its literal sense — “to pursue [the enemy in battle].” But it may be that the meaning of the phrase kol yotzei tzava is “all who go forth to be assembled in the congregation,” because the young men do not take part in such an assembly of the people, and every gathering of the people is called tzava, as in ‘[litzvo] tzava’ (to be counted among the host) for the work of the Tent of Meeting;20Further, 8:24. he shall return from the ‘tz’va’ of the work;21Ibid., Verse 25. with the mirrors of the ‘tzov’oth’ that ‘tzav’u’ (women that congregated together);22Exodus 38:8. and similarly ‘tz’va’ (the host of) heaven;23Deuteronomy 4:19. and all ‘tz’va’am’ (their host) I commanded.24Isaiah 45:12. Therefore Scripture explains when speaking of men of war: ‘mi’tz’va’ (from the host of) the war;25Further, 31:14. and the number of them reckoned ‘bi’tz’va’ (by the host) for the war.26I Chronicles 7: 40. Scripture states here all that go forth to the host, similar to what it says, all that went out of the gate of the city.27Genesis 34: 24. Ramban’s meaning is that the intention of the verses is: “all that are able to go,” even if they have not actually gone. It states, ye shall number them by their ‘hosts’ [in the plural], because they consisted of many hosts, since each and every tribe was a great host.
But as for Rashi’s expression when he wrote — “This is as one says to an executioner: ‘take that man’s head’” — it is not clear to me why the Sages should interpret the verse in this derogatory manner. If it is because [the people counted here] died in the desert [therefore He said, take the head …] whereas in the case of the tribe of Levi He said Number the children of Levi,28Further, 3: 15. since they were not included in the decree [that they die in the desert] — [this cannot be so], for in the second census [taken] of those who were to come into the Land it also uses the same [expression], take the head of all the congregation of the children of Israel!29Ibid., 26:2. But in the Agadah (homiletic exposition) of Vayikra Rabbah30I did not find it there. But a similar exposition is found in Bamidbar Rabbah 1:7. See my Hebrew commentary p. 196, Note 26 for the full text. the Rabbis explain it as an expression of praise [for the people]: “The term se’u always means ‘greatness’, as it is written, Pharaoh ‘yisa’ (shall lift up) thy head, and restore thee unto thy office.31Genesis 40:13. Said the Holy One, blessed be He, to Israel: ‘I have given you an exaltation of the head, and I have likened you to Myself. Just as I am exalted above all inhabitants of the world, as it is said, Thine is the kingdom, O Eternal, and Thou art exalted as head above all,32I Chronicles 29:11. so to you also I have given an elevation of the head, as it is said, Lift up the head of all the congregation of the children of Israel, thus fulfilling that which is said, And He hath lifted up a horn for His people,33Psalms 148:14. and similarly it is said, and the Eternal thy G-d will set thee on high above all the nations of the earth. ' ”34Deuteronomy 28:1.
I have found further in Bamidbar Sinai Rabbah that the Sages have stated:35Bamidbar Rabbah 1:9. “Said Rabbi Pinchas in the name of Rabbi Idi: What is [the meaning of] that which is written at the beginning of this book, ‘se’u eth rosh’ (take the head)? It does not say: ‘lift up the head,’ ‘elevate the head,’ but it says se’u eth rosh, like a man who says to an executioner: ‘take this man’s head’. Here He gave a hint to Moses, se’u eth rosh, meaning that if the people are worthy they will become exalted, just as it is written, Pharaoh ‘yisa’ (shall lift up) thy head, and restore thee unto thy office;31Genesis 40:13. but if they are not worthy, they will all die, just as it is written, Pharaoh ‘yisa’ (shall lift up) thy head from off thee, and shall hang thee on a tree. ”36Genesis 40:19. Thus the expression [se’u eth rosh which, as Rashi explained, is used in connection with execution] is to be interpreted according to the intention [of the speaker] — in a good way for those who are good.37See Psalms 125:4: Do good, O Eternal, unto the good. Thus Pharaoh said unto the chief executioner: “Take the chief butler and lift up his head and restore him to his office.” But as for such as turn aside unto their crooked ways, the Eternal will lead them away with the workers of iniquity (ibid., Verse 5). Thus in the case of the chief baker, the same expression signified his downfall. In short, the expression of the Midrash which Rashi quoted can be interpreted either for bad or for good. And since it is an expression of elevation etc. And since it is [also] an expression of elevation, and it is used [here] in the first census, He used the same expression in the second census.29Ibid., 26:2.
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