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Official Answer:
There is a singular subject which requires a singular verb: the philosophy…has. (A), (B), and (C) contain the verb have. The only difference between (D) and (E) is the use of the words amount and number. Amount is used to refer to items that cannot be counted. Since biographies can be counted, number is correct. (E) is correct.
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The philosophy of the Founding Fathers, the political and military leaders who led the country to its independence, have been the subject of an increasing amount of biographies.

a) have been the subject of an increasing amount
b) have been the subject of an increasing number
c) have subjected an increasing number
d) has been the subject of an increasing amount
e) has been the subject of an increasing number --> correct
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The philosophy of the Founding Fathers,the political and military leaders who led the country to its independence, have been the subject of an increasing amount of biographies.

The Philosophy (So singular)
a) have been the subject of an increasing amount
b) have been the subject of an increasing number
c) have subjected an increasing number

d) has been the subject of an increasing amount Wrong: Amount is used for non countable nouns - water, air, garbage, etc.as the biographies can be counted, correct use will be "number"
e) has been the subject of an increasing number Correct
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IMO E ,

has is used correctly for singular subject philosophy

and number is used correctly

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