binomial formula

简明释义

1. 二项式定理; 2. 二项展开式;

英英释义

例句

1.Students often struggle with the binomial formula 二项式公式 during their first year of calculus.

学生们在微积分的第一年通常会对 binomial formula 二项式公式 感到困惑。

2.The binomial formula 二项式公式 is essential for solving problems involving combinations.

解决涉及组合的问题时，binomial formula 二项式公式 是必不可少的。

3.The binomial formula 二项式公式 allows us to calculate probabilities in binomial distributions.

使用 binomial formula 二项式公式 可以计算二项分布中的概率。

4.Using the binomial formula 二项式公式, we can derive the coefficients for the expansion of (x + y)^5.

通过使用 binomial formula 二项式公式，我们可以推导出 (x + y)^5 展开的系数。

5.In algebra, the binomial formula 二项式公式 is used to expand expressions like (a + b)^n.

在代数中，binomial formula 二项式公式 用于展开像 (a + b)^n 的表达式。

作文

The concept of the binomial formula is a fundamental aspect of algebra that plays a crucial role in various fields, including mathematics, statistics, and even computer science. At its core, the binomial formula provides a way to expand expressions that are raised to a power. This expansion is particularly useful when dealing with polynomial equations and can simplify complex calculations significantly.To understand the binomial formula, we first need to look at what a binomial is. A binomial is simply a mathematical expression that contains two terms, such as (a + b). The binomial formula, often referred to as the Binomial Theorem, states that for any positive integer n, the expansion of (a + b)^n can be expressed as the sum of terms in the form of C(n, k) * a^(n-k) * b^k, where k ranges from 0 to n, and C(n, k) is the binomial coefficient, which represents the number of ways to choose k elements from a set of n elements.The beauty of the binomial formula lies in its ability to provide a systematic method for expanding these expressions without the need for tedious multiplication. For example, if we want to expand (x + y)^3, we can apply the binomial formula to obtain:(x + y)^3 = C(3, 0)x^3y^0 + C(3, 1)x^2y^1 + C(3, 2)x^1y^2 + C(3, 3)x^0y^3= 1*x^3 + 3*x^2*y + 3*x*y^2 + 1*y^3= x^3 + 3x^2y + 3xy^2 + y^3.This expansion shows how each term is derived from the binomial formula, illustrating the coefficients that arise from the binomial coefficients C(3, k).In practical applications, the binomial formula is invaluable. For instance, in probability theory, it helps in calculating the probabilities of different outcomes in binomial experiments, where there are two possible outcomes, such as success or failure. Understanding the binomial formula allows researchers and statisticians to make predictions and analyze data more effectively.Moreover, the binomial formula also has connections to combinatorics, where it aids in counting combinations and arrangements. The coefficients in the binomial formula correspond to the number of ways to choose items from a larger set, making it a powerful tool in solving problems related to counting.In conclusion, the binomial formula is not just a theoretical concept; it is a practical tool that enhances our understanding of algebra and its applications in real-world scenarios. Whether you are a student learning the basics of algebra or a professional working with statistical data, mastering the binomial formula can significantly improve your problem-solving skills and analytical capabilities. As we continue to explore the vast landscape of mathematics, the binomial formula remains a key element that connects various disciplines and enriches our knowledge of the world around us.

二项式公式是代数中的一个基本概念，在数学、统计学甚至计算机科学等多个领域中发挥着重要作用。二项式公式的核心是提供了一种扩展提升幂次的表达式的方法。这种扩展在处理多项式方程时特别有用，可以显著简化复杂的计算。要理解二项式公式，首先需要了解什么是二项式。二项式是一个包含两个项的数学表达式，例如(a + b)。二项式公式，通常被称为二项定理，指出对于任何正整数n，(a + b)^n的展开可以表示为C(n, k) * a^(n-k) * b^k形式的各项之和，其中k从0到n变化，而C(n, k)是二项系数，表示从n个元素中选择k个元素的方式数。二项式公式的美在于它能够提供一种系统的方法来扩展这些表达式，而无需繁琐的乘法。例如，如果我们想扩展(x + y)^3，我们可以应用二项式公式得到：(x + y)^3 = C(3, 0)x^3y^0 + C(3, 1)x^2y^1 + C(3, 2)x^1y^2 + C(3, 3)x^0y^3= 1*x^3 + 3*x^2*y + 3*x*y^2 + 1*y^3= x^3 + 3x^2y + 3xy^2 + y^3。这个展开展示了每一项是如何根据二项式公式推导出来的，说明了来自二项系数C(3, k)的系数。在实际应用中，二项式公式是非常宝贵的。例如，在概率论中，它帮助计算二项实验中不同结果的概率，这些实验只有两种可能的结果，如成功或失败。理解二项式公式使研究人员和统计学家能够更有效地进行预测和分析数据。此外，二项式公式还与组合数学有关，帮助计算组合和排列。二项式公式中的系数对应于从较大集合中选择项目的方式数，使其成为解决与计数相关问题的强大工具。总之，二项式公式不仅仅是一个理论概念；它是一个实用工具，增强了我们对代数及其在现实世界场景中的应用的理解。无论你是学习代数基础的学生，还是处理统计数据的专业人士，掌握二项式公式都能显著提高你的问题解决能力和分析能力。在我们继续探索数学的广阔领域时，二项式公式仍然是一个连接各个学科并丰富我们对周围世界知识的关键要素。