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Take n = 3 and (i1,i2,i3) = (5,6,7)
5,6,7: i1*i2 + i2*i3 + i3*i1 = 30 + 42 + 35
5,7,6: i1*i2 + i2*i3 + i3*i1 = 35 + 42 + 30
6,5,7: i1*i2 + i2*i3 + i3*i1 = 30 + 35 + 42
They are all the same so divide by n! to get the average.