## Understanding the Binomial Expansion of (a + b)^12

The binomial theorem provides a powerful formula for expanding expressions of the form (a + b)^n, where 'n' is a non-negative integer. This article focuses on the expansion of (a + b)^12, demonstrating the application of the binomial theorem and its various aspects.

### The Binomial Theorem

The binomial theorem states that for any non-negative integer 'n', the following equation holds:

**(a + b)^n = ∑_(k=0)^n (n choose k) a^(n-k) b^k**

Where:

**(n choose k)**represents the binomial coefficient, calculated as n! / (k! * (n-k)!).**k**is an integer ranging from 0 to n.

### Expanding (a + b)^12

To expand (a + b)^12, we apply the binomial theorem with n = 12. This results in the following expansion:

**(a + b)^12 = ∑_(k=0)^12 (12 choose k) a^(12-k) b^k**

Let's break down the expansion term by term:

**k = 0:**(12 choose 0) a^12 b^0 = a^12**k = 1:**(12 choose 1) a^11 b^1 = 12a^11 b**k = 2:**(12 choose 2) a^10 b^2 = 66a^10 b^2**k = 3:**(12 choose 3) a^9 b^3 = 220a^9 b^3**k = 4:**(12 choose 4) a^8 b^4 = 495a^8 b^4**k = 5:**(12 choose 5) a^7 b^5 = 792a^7 b^5**k = 6:**(12 choose 6) a^6 b^6 = 924a^6 b^6**k = 7:**(12 choose 7) a^5 b^7 = 792a^5 b^7**k = 8:**(12 choose 8) a^4 b^8 = 495a^4 b^8**k = 9:**(12 choose 9) a^3 b^9 = 220a^3 b^9**k = 10:**(12 choose 10) a^2 b^10 = 66a^2 b^10**k = 11:**(12 choose 11) a^1 b^11 = 12a b^11**k = 12:**(12 choose 12) a^0 b^12 = b^12

Therefore, the complete expansion of (a + b)^12 is:

**(a + b)^12 = a^12 + 12a^11 b + 66a^10 b^2 + 220a^9 b^3 + 495a^8 b^4 + 792a^7 b^5 + 924a^6 b^6 + 792a^5 b^7 + 495a^4 b^8 + 220a^3 b^9 + 66a^2 b^10 + 12a b^11 + b^12**

### Key Observations:

**Symmetry:**Notice the symmetrical pattern in the coefficients. The coefficients for terms equidistant from the beginning and end of the expansion are the same.**Pascal's Triangle:**The binomial coefficients can be conveniently obtained from Pascal's Triangle, where each entry is the sum of the two numbers directly above it.**Applications:**The binomial theorem finds wide applications in various fields like probability, statistics, and calculus. It allows us to express complex expressions in a more manageable form.

By understanding the binomial theorem and its application, we can efficiently expand expressions like (a + b)^12 and leverage its power in various mathematical contexts.