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The expansion is given by $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$

The binomial coefficient $\binom{n}{k}$ is defined as $\frac{n!}{k!(n-k)!}$

The expansion is $(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots$

The binomial series for $(1 + x)^\alpha$, where $\alpha$ is any real number, is given by $(1 + x)^\alpha = \sum_{k=0}^{\infty} \binom{\alpha}{k} x^k$

Each row in Pascal's triangle represents the coefficients of the expanded form of $(a + b)^n$ for $n = 0, 1, 2, \dots$

For real or complex number $z$ and a non-negative integer $k$, the generalized binomial coefficient is $\binom{z}{k} = \frac{\Gamma(z+1)}{\Gamma(k+1)\Gamma(z-k+1)}$

The expansion of $(1-x)^{-n}$ is given by $(1-x)^{-n} = \sum_{k=0}^{\infty} \binom{n+k-1}{k} x^k$

$\binom{n}{k}$ can be interpreted as the number of ways to choose a subset of $k$ elements from a set of $n$ distinct elements

Binomial coefficients are used in the binomial probability distribution, which models the number of successes in a fixed number of independent Bernoulli trials.