## Understanding the Binomial Theorem: (a + b)^10

The expression (a + b)^10 might seem daunting at first glance, but it can be expanded using the **Binomial Theorem**. This powerful theorem allows us to expand any binomial raised to a positive integer power.

### The Binomial Theorem Formula

The general formula for the Binomial Theorem is:

**(a + b)^n = ∑ (n choose k) * a^(n-k) * b^k**

where:

**n**is the power to which the binomial is raised**k**is an integer ranging from 0 to n**(n choose k)**is the binomial coefficient, calculated as n! / (k! * (n-k)!), which represents the number of ways to choose k items from a set of n items.

### Expanding (a + b)^10

To expand (a + b)^10 using the Binomial Theorem, we need to calculate the terms for each value of k from 0 to 10.

**Let's break it down:**

**k = 0:**(10 choose 0) * a^(10-0) * b^0 = 1 * a^10 * 1 =**a^10****k = 1:**(10 choose 1) * a^(10-1) * b^1 = 10 * a^9 * b =**10a^9b****k = 2:**(10 choose 2) * a^(10-2) * b^2 = 45 * a^8 * b^2 =**45a^8b^2****k = 3:**(10 choose 3) * a^(10-3) * b^3 = 120 * a^7 * b^3 =**120a^7b^3****k = 4:**(10 choose 4) * a^(10-4) * b^4 = 210 * a^6 * b^4 =**210a^6b^4****k = 5:**(10 choose 5) * a^(10-5) * b^5 = 252 * a^5 * b^5 =**252a^5b^5****k = 6:**(10 choose 6) * a^(10-6) * b^6 = 210 * a^4 * b^6 =**210a^4b^6****k = 7:**(10 choose 7) * a^(10-7) * b^7 = 120 * a^3 * b^7 =**120a^3b^7****k = 8:**(10 choose 8) * a^(10-8) * b^8 = 45 * a^2 * b^8 =**45a^2b^8****k = 9:**(10 choose 9) * a^(10-9) * b^9 = 10 * a^1 * b^9 =**10ab^9****k = 10:**(10 choose 10) * a^(10-10) * b^10 = 1 * a^0 * b^10 =**b^10**

**Therefore, the expanded form of (a + b)^10 is:**

**a^10 + 10a^9b + 45a^8b^2 + 120a^7b^3 + 210a^6b^4 + 252a^5b^5 + 210a^4b^6 + 120a^3b^7 + 45a^2b^8 + 10ab^9 + b^10**

### Pascal's Triangle

A handy tool for calculating binomial coefficients is **Pascal's Triangle**. Each row of Pascal's Triangle represents the coefficients of the expanded binomial for a given power. The numbers in the triangle are formed by adding the two numbers directly above them.

**Row 0:** 1
**Row 1:** 1 1
**Row 2:** 1 2 1
**Row 3:** 1 3 3 1
**Row 4:** 1 4 6 4 1
**Row 5:** 1 5 10 10 5 1
**...**
**Row 10:** 1 10 45 120 210 252 210 120 45 10 1

Notice that the coefficients in the expansion of (a + b)^10 match the numbers in the 10th row of Pascal's Triangle.

### Conclusion

The Binomial Theorem provides a systematic way to expand binomials raised to any power. Understanding the theorem and its applications is crucial in various fields like algebra, calculus, and probability. While the expansion of (a + b)^10 might seem complex, using the formula and Pascal's Triangle can make the process much easier.