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A binomial coefficient, denoted as \(_{n} C_{k}\), represents the number of ways to choose \(k\) objects from \(n\) objects without considering the order. This concept is a fundamental part of combinatorics. The formula to calculate a binomial coefficient is: \(_{n} C_{k} = \frac{n!}{k!(n-k)!}\). Here, \(n!\) (read as 'n factorial') is the product of all positive integers up to \(n\). So for example, \(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120\). This formula helps us count combinations efficiently.
Binomial coefficients have practical applications in probability, statistics, and algebra. They are involved in expanding expressions using the binomial theorem and in calculating probabilities in binomial distributions.

The binomial theorem provides a way to expand expressions of the form \( (a + b)^n \). According to the theorem: \[ (a + b)^n = \sum_{k=0}^{n}{} _{n} C_{k} a^{n-k} b^k \]. When both \(a\) and \(b\) are set to 1, the expression simplifies to \(2^n\). This means the sum of all binomial coefficients for a given \(n\) is \(2^n\). For our example, with \(n = 100\), we have: \[ _{100} C_{0} + _{100} C_{1} + \cdots + _{100} C_{100} = 2^{100} \]. The binomial theorem is useful in algebra for expanding polynomials and has extensive applications in calculus and higher-level mathematics.

Combinatorics is the branch of mathematics dealing with combinations, permutations, and counting. It involves studying how objects can be arranged or combined under specific rules. Key concepts include:

• Combinations: Selecting items where order does not matter (e.g., binomial coefficients).
• Permutations: Arranging items where order matters.
• Factorials: Used heavily in both combinations and permutations to calculate the total number of ways objects can be ordered or selected.

In combinatorics, binomial coefficients often play a significant role. For example, the sum of binomial coefficients is connected to different counting problems and probability scenarios. Understanding these principles helps solve complex problems in mathematics, such as determining probabilities of certain events or optimizing arrangements.