Problem Solving

If, for all positive integer values of n, P(n) is defined as the sum of the smallest n prime numbers, then which of the following quantities are odd integers?

I. P(10)
II. P(P(10))
III. P(P(P(10)))

(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III

I need help with the solution to the above question.
My approach was:
e=even prime integer
o=odd prime integer

P(10) = e+o+o+o+o+o+o+o+o+o
P(10) = (e+o)+(o+o)+(o+o)+(o+o)+(o+o)
RULE: even+even = even
odd+even = odd
odd+odd = even
Using the above rule-
P(10) = (o)+(e)+(e)+(e)+(e)
P(10) = (o)+[(e)+(e)]+[(e)+(e)]
P(10) = (o)+[e]+[e]
P(10) = odd prime integer

so p(10) is odd

Now for solving,
P(P(10)) = (P(10)+P(10)+P(10)+P(10)+P(10)+P(10)+P(10)+P(10)+P(10)+P(10))
P(P(10)) = (o+o+o+o+o+o+o+o+o+o)
P(P(10)) = even prime integer

Now for solving,
P(P(P(10))) = (P(P(10))+P(P(10))+P(P(10))+P(P(10))+P(P(10))+P(P(10))+P(P(10))+P(P(10))+P(P(10))+P(P(10)))
P(P(P(10))) = (e+e+e+e+e+e+e+e+e+e)
P(P(P(10))) = even prime integer

So per my approach I get "(A)" as the answer but the correct answer is "(C)"
I believe there is a flaw in my approach.
Would someone please help me understand the question better.

Thanks...