Even/Odd properties of integer algebraic expressions

Analyze each parity expression based on the fixed inventory of three even (E) and two odd (O) integers[cite: 3105, 3106]:

Statement 1: p + q + r - s - t. In algebra, parity is invariant under addition and subtraction (i.e., A - B has the same parity as A + B). Therefore, the expression simplifies to the sum of all five numbers: p + q + r + s + t. Sum = E + E + E + O + O = Even + Even = Even. Thus, it is definitely even. Statement 1 is correct.

Statement 2: 2p + q + 2r - 2s + t. Any term multiplied by 2 becomes strictly even regardless of its origin (2p, 2r, 2s \rightarrow E). The expression collapses to: Even + q + Even - Even + t = Even + (q + t). Since q and t are part of the original five-integer pool, let's look at the worst cases for the leftover elements (q, t):

Since the result depends on which specific variables hold the odd values, it is not definitely odd.

Self-Correction on Statement 2 analysis: Let's re-verify if the question parameters restrict the allocation. If q and t can be either even or odd depending on the permutation, then it is not definitely odd. Thus, only Statement 1 holds with absolute certainty.

Answer: (a).