Newton's binomial

Newton's binomial theorem states that for any natural number $n$, (a + b)^n = inom{n}{0}a^n b^0 + inom{n}{1}a^{n-1}b^1 + inom{n}{2}a^{n-2}b^2 + \\ \\ ... + inom{n}{n}a^0b^n, where $\binom{n}{k}$ is the binomial coefficient, representing the number of ways to choose $k$ objects from $n$.

Important properties

• The expansion has $(n+1)$ terms.

• The exponents of $a$ decrease from $n$ to $0$, while those of $b$ increase from $0$ to $n$.

• The coefficients are given by binomial coefficients: $\binom{n}{k}$.

• The sum of the exponents in each term is always $n$.