Has the length of a man's life already been determined in advance, even before a person is born?
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Yes, it is predetermined. Right when a person is a foetus in the womb, his span of life is already determined based on his past karma. Here's what chapter 32 of the Uttara Kanda of the Garuda Purana says:
O Tarksya, thus, man is born in the mortal world as a result of his own actions. The five characteristics, viz. the span of life, action, wealth, learning and death are created in men even as they are in the womb. The life is born as they are in the womb. The life is born as a result of action; it is dissolved as a result of action. Happiness or misery, fear or welfare are the results of actions. The foetus lies in the womb with the feet up and the face downwards. The foetus lies in the womb with the feet up and the face downwards. The life takes its birth with its association brought about by its own actions.
This excerpt from the Bhumi Kanda of the Padma Purana says the same thing:
Of the fetus in the womb these five are seen: (span of) life, acts, wealth, learning and death. As the doer does (i.e. fashions different images from) a lump of clay, similarly acts that are performed grasp the doer. A being reached the state of a god or a human being or a beast or a bird or a lower animal or an immobile thing due to his acts only. He always enjoys in accordance with what is done by himself. He, having acquired the bed (in the form) of the womb enjoys pleasure or pain ordained by himself (and due to acts) in the former body (i.e. existence).
You can read the rest of the Padma Purana here.
By the way, it might seem strange that even a person's actions would be fixed from the time they're just a foetus, but it doesn't mean that people don't act and make choices. Rather, as the Brahma Sutras discuss, people genuinely act, but it is still the case that Brahman causes them to act; see my answer here.
Well, the answer needs a proper understanding of the concept of Karma. Hindu philosophy divides Karma into - Prarabdha, Sanchita and Agami.
Sanchita Karma is the store house of all past Karmas. It is our Causal body/Karana sharira.
Prarabdha Karma refers to those Karmas which has started to give results. A person gets a birth, takes on a body based on this Prarabdha Karma. Hence, even at the time of birth, the fruits of actions which a person is supposed to face is already decided including the lifeline of a person. But, this is not static. This does not mean there is no role for free Will.
The fruits of Prarabdha Karmas manifest as various situations that a person faces, as various people that a person meets through rina, and as various mental impressions that govern one's decisions. But, inspite of all this, a person has a "Free-Will".
The exertion of free will is called "Agami Karma" and it may give fruit in this life itself or may be stored as Sanchita to give fruits in future lives.
Therefore, though, a general lifeline of a person is decided right at birth, by free will, a person can elongate or shorten it. Example, Markandeya had very short life, but his devotion to Shiva, which was exertion of free will, resulted in Shiva granting him immortality.
Similarly, a person who performs too much adharma and cause sorrow to others, may also have his life shortened.
Hope this helps.