binomial coefficient

简明释义

二项式系数;

英英释义

例句

1.When expanding a binomial expression, the coefficients are determined by the binomial coefficient 二项式系数.

在展开一个二项式表达式时，系数由 binomial coefficient 二项式系数 决定。

2.The formula for calculating the binomial coefficient 二项式系数 is given by n! / (k!(n-k)!).

计算 binomial coefficient 二项式系数 的公式为 n! / (k!(n-k)!)。

3.In combinatorics, the binomial coefficient 二项式系数 represents the number of ways to choose k elements from a set of n elements.

在组合数学中，binomial coefficient 二项式系数 表示从 n 个元素的集合中选择 k 个元素的方式数量。

4.The binomial coefficient 二项式系数 can be visualized using Pascal's triangle.

可以通过帕斯卡三角形来可视化 binomial coefficient 二项式系数。

5.In probability theory, the binomial coefficient 二项式系数 is used to calculate the likelihood of a certain number of successes in a series of trials.

在概率论中，binomial coefficient 二项式系数 用于计算在一系列试验中某个成功次数的可能性。

作文

The concept of the binomial coefficient is fundamental in combinatorics, a branch of mathematics that deals with counting, arrangement, and combination of objects. In essence, the binomial coefficient represents the number of ways to choose a subset of items from a larger set, without regard to the order of selection. It is denoted as C(n, k) or sometimes as n choose k, where n is the total number of items, and k is the number of items to choose. The formula for calculating the binomial coefficient is given by C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial, which is the product of all positive integers up to a given number.Understanding the binomial coefficient is crucial for various applications, including probability theory, statistics, and algebra. For instance, in probability, the binomial coefficient is used to determine the likelihood of a certain number of successes in a series of independent experiments, each with two possible outcomes. This application is particularly useful in fields such as genetics, finance, and quality control, where one might want to calculate the probabilities of different outcomes based on a fixed number of trials.Moreover, the binomial coefficient plays a significant role in the expansion of binomial expressions, as described by the Binomial Theorem. This theorem states that (a + b)^n can be expanded into a sum involving the binomial coefficient. Specifically, it can be expressed as:(a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n.This means that each term in the expansion is multiplied by a corresponding binomial coefficient, which indicates how many ways we can select k instances of b from n total instances of either a or b. This theorem not only highlights the importance of the binomial coefficient in algebra but also provides a method to compute powers of binomials efficiently.In addition to its theoretical significance, the binomial coefficient has practical applications in computer science, particularly in algorithms related to combinatorial problems. For example, when designing algorithms to solve problems like the traveling salesman problem or generating combinations of elements from a set, understanding how to calculate the binomial coefficient can lead to more efficient solutions.As we delve deeper into the world of mathematics, it becomes evident that the binomial coefficient is not just an abstract concept but a tool that has far-reaching implications in various fields. Whether it's analyzing data trends, predicting outcomes, or solving complex problems, the binomial coefficient serves as a bridge connecting different areas of study. Its versatility and utility make it an essential topic for anyone interested in mathematics or its applications.In conclusion, mastering the binomial coefficient opens up a wealth of knowledge and applications in both theoretical and practical contexts. By understanding its definition, formula, and applications, one can appreciate the elegance and power of this mathematical concept. As you continue your journey in mathematics, keep the binomial coefficient in mind, as it will undoubtedly enhance your problem-solving toolkit and provide insights into the fascinating world of combinatorial mathematics.

“二项式系数”是组合数学中的一个基本概念，组合数学是研究对象的计数、排列和组合的数学分支。简单来说，“二项式系数”表示从较大集合中选择子集的方式数量，而不考虑选择的顺序。它表示为C(n, k)，有时也称为n选k，其中n是总物品数量，k是要选择的物品数量。计算“二项式系数”的公式为C(n, k) = n! / (k!(n-k)!)，其中'!'表示阶乘，即所有正整数的乘积，直到给定数字。理解“二项式系数”对各种应用至关重要，包括概率论、统计学和代数。例如，在概率中，“二项式系数”用于确定在一系列独立实验中成功的次数的可能性，每次实验都有两个可能的结果。这种应用在遗传学、金融和质量控制等领域特别有用，在这些领域中，人们可能希望根据固定的试验次数计算不同结果的概率。此外，“二项式系数”在二项式表达式的展开中也发挥着重要作用，如二项式定理所描述的那样。该定理指出(a + b)^n可以展开为涉及“二项式系数”的和。具体来说，它可以表示为：(a + b)^n = Σ (C(n, k) * a^(n-k) * b^k)，对于k = 0到n。这意味着展开中的每一项都乘以相应的“二项式系数”，表示我们可以从n个总实例中选择k个b的方式数量。该定理不仅突显了“二项式系数”在代数中的重要性，还提供了一种有效计算二项式幂的方法。除了其理论意义外，“二项式系数”在计算机科学中也有实际应用，特别是在与组合问题相关的算法中。例如，在设计解决旅行推销员问题或生成集合元素组合的算法时，理解如何计算“二项式系数”可以导致更高效的解决方案。当我们深入探索数学世界时，显而易见的是，“二项式系数”不仅是一个抽象概念，而是一个具有广泛影响力的工具，应用于各个领域。无论是分析数据趋势、预测结果，还是解决复杂问题，“二项式系数”都作为连接不同研究领域的桥梁。它的多功能性和实用性使其成为任何对数学或其应用感兴趣的人必备的主题。总之，掌握“二项式系数”开启了理论和实践背景下丰富的知识和应用。通过理解其定义、公式和应用，可以欣赏到这一数学概念的优雅和力量。在继续你在数学中的旅程时，请记住“二项式系数”，因为它无疑会增强你的问题解决工具包，并为你提供对组合数学迷人世界的洞察。