Let a, b, c and d be four positive integers such that a+b+c+d=200. If S=(-1)^{a}+(-1)^{b}+(-1)^{c}+(-1)^{d}, then what is the number of possible values of S?

Explanation

Since a+b+c+d=200 (an even sum), the number of odd integers among them must be even (0, 2, or 4). If 0 are odd, S = 4(1) = 4. If 2 are odd, S = 2(1) + 2(-1) = 0. If 4 are odd, S = 4(-1) = -4. Thus, S can take 3 possible values.

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