`(x + y)^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 x for x, y for y and 5 for n, such that:

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

`(x+y)^5 = x^5+5x^4*y+10x^3*y^2+10x^2*y^3 +5x*y^4+y^5`

Hence, expanding the complex number using binomial theorem yields the simplified result `(x+y)^5 = x^5+5x^4*y+10x^3*y^2+10x^2*y^3 +5x*y^4+y^5.`