All about binomial theorem

The binomial theorem provides a formula for expanding expressions of the form $(a+b)^n$ for any positive integer $n$. It simplifies the process of multiplying a binomial by itself many times.

The general formula for the binomial theorem is:

$$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

Where:

• $n$ is a non-negative integer.
• $k$ is an integer from 0 to $n$.
• $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$.

For example, let's expand $(x+y)^3$:

$$\begin{align*} (x+y)^3 &= \binom{3}{0} x^{3-0} y^0 + \binom{3}{1} x^{3-1} y^1 + \binom{3}{2} x^{3-2} y^2 + \binom{3}{3} x^{3-3} y^3 \\ &= 1 \cdot x^3 \cdot 1 + 3 \cdot x^2 \cdot y + 3 \cdot x^1 \cdot y^2 + 1 \cdot 1 \cdot y^3 \\ &= x^3 + 3x^2y + 3xy^2 + y^3 \end{align*}$$