## Expanding (a + b)^6: The Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions of the form (a + b)^n, where n is a non-negative integer. This theorem allows us to efficiently determine the coefficients of the terms in the expansion.

### Understanding the Binomial Theorem

The binomial theorem states that:

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

where:

**n**is the power of the binomial.**k**ranges from 0 to n.**(n choose k)**represents the binomial coefficient, which is calculated as n!/(k!*(n-k)!).

### Expanding (a + b)^6

To expand (a + b)^6, we can use the binomial theorem by setting n = 6 and applying the formula:

(a + b)^6 = ∑(6 choose k) * a^(6-k) * b^k

Let's calculate each term for k from 0 to 6:

**k = 0:**(6 choose 0) * a^(6-0) * b^0 = 1 * a^6 * 1 =**a^6****k = 1:**(6 choose 1) * a^(6-1) * b^1 = 6 * a^5 * b =**6a^5b****k = 2:**(6 choose 2) * a^(6-2) * b^2 = 15 * a^4 * b^2 =**15a^4b^2****k = 3:**(6 choose 3) * a^(6-3) * b^3 = 20 * a^3 * b^3 =**20a^3b^3****k = 4:**(6 choose 4) * a^(6-4) * b^4 = 15 * a^2 * b^4 =**15a^2b^4****k = 5:**(6 choose 5) * a^(6-5) * b^5 = 6 * a^1 * b^5 =**6ab^5****k = 6:**(6 choose 6) * a^(6-6) * b^6 = 1 * a^0 * b^6 =**b^6**

Therefore, the complete expansion of (a + b)^6 is:

**(a + b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6**

### Key Observations

**Symmetry:**Notice the symmetry in the coefficients. The coefficients are mirrored from the beginning and the end of the expansion.**Pascal's Triangle:**The coefficients of the binomial expansion can be easily found using Pascal's Triangle.**Applications:**The binomial theorem has extensive applications in algebra, calculus, probability, and statistics.

By applying the binomial theorem, we can efficiently expand any expression of the form (a + b)^n, making complex calculations much simpler.