Algebra - Binomial Theorem

Binomial Expression and Expansion: A binomial is an algebraic expression containing two terms, such as $(a + b)$. The Binomial Theorem provides a formula for expanding any positive integer power of a binomial, $(a + b)^n$. In the expansion, the powers of the first term '$a$' decrease from $n$ to $0$, while the powers of the second term '$b$' increase from $0$ to $n$.

Pascal's Triangle (Visual Concept): This is a geometric arrangement of binomial coefficients. It looks like a triangle where the number at the apex is $1$. Each number below is the sum of the two numbers directly above it. Visually, the triangle is perfectly symmetric; the coefficients on the left side are identical to those on the right, which corresponds to the property $^{n}C_r = ^{n}C_{n-r}$.

General Term ($T_{r+1}$): The $(r+1)^{th}$ term in the expansion of $(a + b)^n$ is denoted by $T_{r+1}$. This formula allows us to find any specific term in the expansion without writing out the whole series. Note that to find the $k^{th}$ term, we substitute $r = k-1$.

Total Number of Terms: The number of terms in the expansion of $(a + b)^n$ is always $n + 1$. For example, $(a + b)^2$ expands to $a^2 + 2ab + b^2$, which contains $3$ terms. This is a linear progression where the count of terms is always one more than the index of the power.

Middle Term(s): The position of the middle term depends on whether $n$ is even or odd. If $n$ is even, the expansion has an odd number of terms, resulting in one middle term at the $(\frac{n}{2} + 1)^{th}$ position. If $n$ is odd, the expansion has an even number of terms, resulting in two middle terms at the $(\frac{n+1}{2})^{th}$ and $(\frac{n+3}{2})^{th}$ positions. Visually, these represent the central values in a row of Pascal's Triangle.

Properties of Coefficients: The sum of all binomial coefficients in the expansion of $(1 + x)^n$ is $2^n$ (obtained by setting $x = 1$). Furthermore, the sum of coefficients of odd terms is equal to the sum of coefficients of even terms, both being equal to $2^{n-1}$.

Index Rules: In every term of the expansion of $(a + b)^n$, the sum of the indices (exponents) of $a$ and $b$ is always equal to $n$. For instance, in the expansion of $(x + y)^5$, a term like $x^2y^3$ has a total index sum of $2 + 3 = 5$.