Determine the coefficient of x^3 in the expansion of (1 – x)^5(1 + 1/x)^5

Notice that

(1 - x)⁵ (1 + 1/x)⁵ = ((1 - x) (1 + 1/x))⁵ = (1 - x + 1/x - 1)⁵ = (1/x - x)⁵

Recall the binomial theorem:

Let a = 1/x, b = -x, and n = 5. Then

We get an x ³ term for

2k - 5 = 3   ==>   2k = 8   ==>   k = 4

so that the coefficient would be

no, because the sides are not congruent