How Many People Will Enter Jannah?
Question

In Sahih Al-Bukhari, it is mentioned that ...70,000 of all mankind will enter paradise! The first three generations of Muslims alone probably amount to 70,000. Is there another explanation to this Hadith?
Praise be to Allah.Hadith about people entering Paradise without reckoning 
Perhaps you are referring to the Hadith about the seventy thousand who will enter Paradise without being brought to account, which was reported by the two Shaykhs (Al-Bukhari and Muslim), Ahmad and others from the Prophet (peace and blessings of Allah be upon him). If you study this Hadith and think about it, then in sha Allah the confusion you describe in your question will be resolved.
Imam Al-Bukhari (may Allah have mercy on him) narrated from Ibn `Abbas (may Allah be pleased with them both) that the Prophet (peace and blessings of Allah be upon him) said: “I was shown the nations, and some Prophets passed by with a few followers, and some Prophets passed by with no followers. Then I was shown a great multitude, and I said, ‘What is this? Is this my Ummah?’ It was said, ‘No, this is Moosa and his people.’ It was said, ‘Look at the horizon.’ There I saw a huge multitude filling the horizon. And it was said, ‘Look there, and there, on the horizons of the sky.’ There was a multitude filling the horizons. It was said, ‘This is your Ummah, and of these, seventy thousand will enter Paradise without being brought to account.’” Then he [the Prophet (peace and blessings of Allah be upon him)] went inside, without explaining further. The people started to discuss what he had said, saying, “We are the ones who have believed in Allah and followed His Messenger; we are they” or, “It is our children who were born in Islam whilst we were born in Jahiliyyah.” The Prophet (peace and blessings of Allah be upon him) heard about what was being said, so he came out and said: “They are the ones who do not seek Ruqya, do not believe in bad omens and do not use branding; they put their trust in their Lord…”
What the Hadith means is that there is a group of this Ummah who will enter Paradise without being brought to account, not that the number of inhabitants of Paradise from this Ummah is seventy thousand. The seventy thousand referred to in this Hadith are of a high status in this Ummah because of some special qualities that they have, as mentioned in the Hadith: “They are the ones who do not seek Ruqya, do not believe in bad omens and do not use branding; they put their trust in their Lord…”
Description of those who will enter Paradise without reckoning
The reason why they will enter Paradise without being brought to account and without being punished is stated clearly in another report narrated by Al-Bukhari (may Allah have mercy on him) from Ibn `Abbas, who said: “The Prophet (peace and blessings of Allah be upon him) said: ‘I was shown the nations. One Prophet passed by with a group, another passed by with a small band, another with ten followers, another with five, and another on his own (with no followers). I looked and saw a huge multitude, and I said, “O Jibril, are these my Ummah?” He said “No, but look at the horizon.” So I looked and saw a huge multitude. He said, “These are your Ummah, and these are seventy thousand at their head who will not be brought to account or punished.” I said, “How come?” He said, “They do not use branding, or seek Ruqya, or believe in bad omens; they put their trust in their Lord.”’ ‘Ukhashah ibn Mihsan stood up and said, ‘Pray to Allah to make me one of them!’ He [the Prophet (peace and blessings of Allah be upon him)] said, ‘O Allah, make him one of them.’ Then another man stood up and said, ‘Pray to Allah to make me one of them.’ He said, ‘ ‘Ukashah has beaten you to it.’” (Narrated by Al-Bukhari, 6059)
They are described further in another Hadith, narrated by Sahl ibn Sa`d (may Allah be pleased with him), according to which the Prophet (peace and blessings of Allah be upon him) said: “Seventy thousand or seven hundred thousand [one of the narrators was not sure] of my Ummah will enter Paradise, and the first of them will not enter until the last of them do so. And their faces will be like the moon on the night when it is full.” (Narrated by Al-Bukhari)
Abu Hurayrah (may Allah be pleased with him) said: “I heard the Messenger of Allah (peace and blessings of Allah be upon him) say: ‘A group of my Ummah seventy thousand strong will enter Paradise, with their faces shining like the moon.” (Narrated by Al-Bukhari)
Muslim also reported about them in his Saheeh from Jabir ibn `Abdullah: “… then the believers will be saved and the first group to be saved will have faces like the moon when it is full, [they will be] seventy thousand, who will not be brought to account. Then will come those who shine like the stars in the sky…”
How many people will enter Jannah?
There is good news for all of us Muslims in this Hadith and others. There is more information in a report narrated in the Musnad of Imam Ahmad and the Sunan of At-Tirmidhi and Ibn Majah, where Abu Umamah (may Allah be pleased with him) is reported to have said that the Prophet (peace and blessings of Allah be upon him) said: “My Lord promised me that seventy thousands of my Ummah will enter Paradise without being brought to account or being punished; with each thousand of them will be another seventy thousand, plus three handfuls of the handfuls of my Lord, may He be glorified.” We ask Allah to make us among them. Just imagine: seventy thousand, with each thousand of whom is another seventy thousand – how great will be the total number of people who will enter Paradise without being brought to account? How many will be in each handful “scooped up” by the Merciful, Compassionate, Generous Almighty Lord? We ask Allah to make us among that number.
The second item of good news is that the number of inhabitants of Paradise from this Ummah will be two thirds of the total number. More people will enter Paradise from the Ummah of Muhammad (peace and blessings of Allah be upon him) than from all the previous nations combined. This good news comes from the Prophet (peace and blessings of Allah be upon him) in the Hadith in which he said to his Companions one day, “Would you be pleased if you were a quarter of the people of Paradise?” They said, “Yes.” He said, “Would you be pleased if you were a third of the people of Paradise?” They said, “Yes.” He said, “Would you be pleased if you were a half of the people of Paradise?” They said, “Yes.” He said, “By the One in Whose hand is the soul of Muhammad, I hope that you will be half of the people of Paradise. No one will enter Paradise except a Muslim, and among the people of shirk you are like a white hair on the hide of a black bull, or a black hair on the hide of a red bull.” (Narrated by Al-Bukhari, 6047). The final part of the good news comes in another authentic Hadith, in which he (peace and blessings of Allah be upon him) said: “The people of Paradise are 120 rows, of which eighty are from this Ummah and the other forty from all the other nations.” (Narrated by At-Tirmidhi, 3469; he said this is a sound Hadith). We praise Allah for His blessings and ask Him for His grace and mercy, and to make us dwell in Paradise by His mercy and kindness. May Allah bless our Prophet Muhammad.
And Allah knows best.