## Understanding the Binomial Expansion of (x-y)^5

The binomial theorem allows us to expand expressions of the form (x+y)^n, where n is a positive integer. In this case, we'll explore the expansion of (x-y)^5.

### The Binomial Theorem

The binomial theorem states that:

**(x + y)^n = โ_(k=0)^n (n choose k) * x^(n-k) * y^k**

where:

**(n choose k)**represents the binomial coefficient, calculated as n! / (k! * (n-k)!). It represents the number of ways to choose k items from a set of n items.**โ_(k=0)^n**represents the sum from k=0 to n.

### Expanding (x-y)^5

Applying the binomial theorem to (x-y)^5, we get:

**(x-y)^5 = โ_(k=0)^5 (5 choose k) * x^(5-k) * (-y)^k**

Let's expand this sum term by term:

**k=0:**(5 choose 0) * x^(5-0) * (-y)^0 = 1 * x^5 * 1 =**x^5****k=1:**(5 choose 1) * x^(5-1) * (-y)^1 = 5 * x^4 * (-y) =**-5x^4y****k=2:**(5 choose 2) * x^(5-2) * (-y)^2 = 10 * x^3 * y^2 =**10x^3y^2****k=3:**(5 choose 3) * x^(5-3) * (-y)^3 = 10 * x^2 * (-y^3) =**-10x^2y^3****k=4:**(5 choose 4) * x^(5-4) * (-y)^4 = 5 * x * y^4 =**5xy^4****k=5:**(5 choose 5) * x^(5-5) * (-y)^5 = 1 * 1 * (-y^5) =**-y^5**

Therefore, the complete expansion of (x-y)^5 is:

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

### Key Points

- The binomial expansion of (x-y)^5 has six terms.
- The coefficients of each term are the binomial coefficients, which can be calculated using the formula (n choose k).
- The signs alternate between positive and negative, starting with a positive sign.
- The powers of x decrease from 5 to 0, while the powers of y increase from 0 to 5.

Understanding the binomial theorem and its application to expanding expressions like (x-y)^5 is crucial in various fields, including algebra, calculus, and probability.