binomikerroin

Binomikerroin, often denoted as $\binom{n}{k}$, is a mathematical concept representing the number of ways to choose $k$ items from a set of $n$ distinct items without regard to the order of selection. It is a fundamental concept in combinatorics and probability. The formula for the binomial coefficient is given by $\frac{n!}{k!(n-k)!}$, where $n!$ (n factorial) is the product of all positive integers up to $n$.

The binomial coefficient is crucial in the binomial theorem, which states that $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$. This theorem expands a binomial power into a sum of terms involving binomial coefficients. Binomikerroin also appears in various probability distributions, most notably the binomial distribution, which describes the number of successes in a fixed number of independent Bernoulli trials.

The value of $\binom{n}{k}$ is always a non-negative integer. For instance, $\binom{5}{2}$ represents the number of ways to choose 2 items from a set of 5, which is $\frac{5!}{2!(5-2)!} = \frac{120}{2 \times 6} = 10$. Conventionally, $\binom{n}{0} = 1$ and $\binom{n}{n} = 1$, signifying that there is only one way to choose zero items or all $n$ items. If $k > n$ or $k < 0$, the binomial coefficient is defined as 0.