Use Pascal's triangle to expand the binomial $(x - 5y)^5$.

1. $x^5 - 5x^4(5y)$
2. $\binom{5}{2}x^3(5y)^2 - \binom{5}{3}x^2(5y)^3$
3. $\binom{5}{4}x(5y)^4 - (5y)^5$

Now, simplify each term:

1. $x^5 - 25x^4y$
2. $+ 100x^3y^2 - 250x^2y^3$
3. $+ 500xy^4 - 3125y^5$

Thus, the expanded form is:

$x^5 - 25x^4y + 100x^3y^2 - 250x^2y^3 + 500xy^4 - 3125y^5$

To expand the binomial (x - 5y)⁵ using Pascal's triangle, we use the fifth row of Pascal's triangle and apply the binomial theorem, giving us the expanded form x⁵ - 5x⁴y + 10x³y² - 10x²y³ + 5xy⁴ - y⁵.

Explanation:

To use Pascal's triangle to expand the binomial (x - 5y)⁵, we look at the row of Pascal's triangle corresponding to the exponent, which is 5.

The row for n=5 is 1, 5, 10, 10, 5, 1. We use the binomial theorem to expand the expression, with each term taking coefficients from Pascal's triangle, the variable x decreasing in power, and the variable y increasing in power, all multiplied appropriately.

The first term is x⁵.
The second term is – 5 times x⁴ times y (since the sign alternates).
The third term is 10x³y² (note the square of -5y is 25y²).
The fourth term is – 10x²y³.
The fifth term is 5xy⁴.
The final term is – y⁵.

Combining all the terms gives us the expanded form:

(x - 5y)⁵ = x⁵ - 5x⁴y + 10x³y² - 10x²y³ + 5xy⁴ - y⁵