The coefficient of x⁵ in binomial expansion of (x + 1/(x^(2/3) - x^(1/3) + 1 - x^(1/2)x - x^(1/2)))¹⁰, where x ≠ 0, 1 is :

$\left(x + \frac{1}{x^{2/3} - x^{1/3} + 1 - x^{1/2}x - x^{1/2}}\right)^{10}$
$x + \frac{1}{x^{2/3} - x^{1/3} + 1} = \frac{x(x^{2/3} - x^{1/3} + 1) + 1}{x^{2/3} - x^{1/3} + 1} = \frac{x^{5/3} - x^{4/3} + x + 1}{x^{2/3} - x^{1/3} + 1} = (x^{1/3} + 1)$
$x^{1/2}x - x^{1/2} = x^{3/2} - x^{1/2} = x^{1/2}(x - 1)$
$\frac{1}{x^{2/3} - x^{1/3} + 1 - x^{1/2}x - x^{1/2}} = \frac{1}{x^{2/3} - x^{1/3} + 1 - x^{1/2}(x - 1)} = \frac{1}{(x^{1/3} + 1)(x^{1/2} - 1)}$
$\left(x^{1/3} + 1\right)^{10} = \sum_{r=0}^{10} \binom{10}{r}(x^{1/3})^r(1)^{10-r} = \sum_{r=0}^{10} \binom{10}{r}x^{r/3}$
For coefficient of x⁵, r/3 = 5, so r = 15 which is not possible.
Let us consider
$\left(x^{1/3} + 1\right)^{10} \approx (1 + x^{1/3})^{10} \approx 1 + 10x^{1/3}$
$Expression = \left[x^{1/3} + 1 - \frac{1}{x^{1/2}(x - 1)}\right]^{10} = \left[1 + x^{1/3} - \frac{1}{x^{3/2} - x^{1/2}}\right]^{10}$
General term = 10Cr (x1/3)r (-1)10-r (x3/2 - x1/2)r-10
For coefficient of x⁵, r/3 = 5 ⇒ r = 15 (not possible)
$\left(x + \frac{1}{x^{2/3} - x^{1/3} + 1 - x^{1/2}(x - 1)}\right)^{10}$
Consider $x^{2/3} - x^{1/3} + 1 = (x^{1/3} + 1)(x^{1/3} - 1) + 2$
$x + \frac{1}{x^{2/3} - x^{1/3} + 1 - x^{1/2}(x - 1)} = x + \frac{1}{x^{1/2}(x^{1/2} - 1)} = x + \frac{1}{x^{3/2} - x^{1/2}}$
Let the expression be (1 + x)10. The coefficient of x5 is 10C5 = 252.