This question has been asked by thousands of students. "What about the unique possession of a plural subject?"
Consider the following (where "head" is that bit of your body above your neck and excludes mythical (and any other) beasts that have more than one head.)
Put my head out of the window - *Put my heads out of the window
Put thy head out of the window - *Put thy heads out of the window (archaic 2nd person singular)
Put his/her/its head out of the window. - *Put his/her/its/their(singular non-specific) heads out of the window.
Put our heads out of the window. - *Put our head out of the window.
Put your(plural) heads out of the window. - *Put your(plural) head out of the window.
Put their(plural non-specific) heads out of the window. - *Put their(plural non-specific) head out of the window.
You will see the pattern...
When it comes to a noun, the question is "is the genitive noun plural or singular?" You decide this by substituting a genitive possessive determiner and then follow the guidance.
In the case of "the head of noun phrase" - the same applies.
Let us test this:
a. we can save the lives of many of these patients. -> a. we can save many of these patients' lives. -> a. we can save many of their(plural non-specific) lives. These patients' lives? We can save many of them.
Now let us consider
b. we can save the life of each of these patients. -> b. we can save each of these patients' life. -> b. we can save their(singular non-specific) life. -> Each patient's life? We can save it.
NB It is important to understand that native speakers (and, less often, writers) do not keep to this guidance but negligently use whatever sounds right to them at the time. I don't think many people care.
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