`(2x - 5y)^5` Use the Binomial Theorem to expand and simplify the expression.

You need to use the binomial formula, such that:

`(x+y)^n = sum_(k=0)^n ((n),(k)) x^(n-k) y^k`

You need to replace 2x for x, 5y for y and 5 for n, such that:

`(2x - 5y)^5 = 5C0 (2x)^5+5C1 (2x)^4*(-5y)^1+5C2 (2x)^3*(-5y)^2+5C3 (2x)^2*(-5y)^3 + 5C4 2x*(-5y)^4 + 5C5 (-5y)^5`

By definition, nC0 = nCn = 1, hence `5C0 = 5C5 = 1.`

By definition nC1 = nC(n-1) = n, hence `5C1 = 5C4 = 5.`

By definition `nC2 = (n(n-1))/2` , hence `5C2 = 5C3 = 10.`

`(2x - 5y)^5 = 32x^5-400x^4*y+2000x^3*y^2-5000x^2*y^3 + 6250x*y^4 -3125y^5`

Hence, expanding the complex number using binomial theorem yields the simplified result `(2x - 5y)^5 = 32x^5-400x^4*y+2000x^3*y^2-5000x^2*y^3 + 6250x*y^4 -3125y^5.`