Pascal’s Triangle: Binomial Expansion and Coefficients | 帕斯卡三角形：二项式展开与系数

📚 Pascal’s Triangle: Binomial Expansion and Coefficients | 帕斯卡三角形：二项式展开与系数

In Edexcel A Level Mathematics, Pascal’s triangle is much more than a neat number pattern. It gives you the coefficients in binomial expansions such as (a + b)ⁿ, and it underpins binomial probability calculations. Mastering the triangle and the related nCr notation will make many Pure and Statistics questions faster and more accurate.

在爱德思 A Level 数学中，帕斯卡三角形远不止是一个整齐的数字图案。它为 (a + b)ⁿ 等二项式展开提供系数，并且是二项概率计算的基础。掌握帕斯卡三角形以及相关的 nCr 记号，可以让你在纯数学和统计题中做得更快、更准确。

1. What is Pascal’s Triangle? | 什么是帕斯卡三角形？

Pascal’s triangle is an infinite triangular array of numbers. Each row starts and ends with 1, and every internal number is the sum of the two numbers directly above it. For positive integer n, the nth row of the triangle gives the coefficients of (1 + x)ⁿ.

帕斯卡三角形是一个无限的三角形数字阵列。每一行以 1 开始并以 1 结束，每一个内部数字都是其正上方两个数字之和。对于正整数 n，三角形的第 n 行给出 (1 + x)ⁿ 的系数。

Although named after Blaise Pascal, the same pattern was known earlier in China, India, and Persia. In A Level exams you will usually use it to expand binomial expressions and to find coefficients quickly.

尽管该三角形以布莱兹·帕斯卡的名字命名，但中国、印度和波斯早在之前就已知道这一规律。在 A Level 考试中，你通常会用它对二项式进行展开并快速求出系数。

2. Constructing the Triangle | 构造帕斯卡三角形

To build the triangle, start with a single 1 at the top. Under it, write two 1s. For every new row, place 1 at both ends, then fill each internal position by adding the two numbers above-left and above-right.

要构造三角形，从顶部的一个 1 开始。在它下面写出两个 1。对于每一新行，两端都放 1，然后将左上方和右上方的两个数字相加，填入每个内部位置。

The first six rows are shown below:

前六行如下所示：

Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5

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