Explanation

With common denominator xyz(x−y)(y−z)(z−x), the numerator sums to yz(y−z)+zx(z−x)+xy(x−y) = (x−y)(y−z)(z−x), so the whole sum collapses to 1/(xyz). As a check, x=1,y=2,z=3 gives 1/2 − 1/2 + 1/6 = 1/6. The keyed option is C.