Pascal’s Triangle | 帕斯卡三角形

📚 Pascal’s Triangle | 帕斯卡三角形

Pascal’s triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it. It appears in algebra, combinatorics, and probability, and is named after the French mathematician Blaise Pascal, although it was known centuries earlier in China and Persia.

帕斯卡三角形是一个数字三角形阵列，其中每个数字等于它上方两数之和。它出现在代数、组合学和概率中，以法国数学家布莱兹·帕斯卡命名，尽管几个世纪前中国和波斯就已知道这一图形。

1. Introduction to Pascal’s Triangle | 帕斯卡三角形简介

Pascal’s triangle is built starting with a top row containing a single 1. Each subsequent row begins and ends with 1, and every interior entry is obtained by adding the two entries above it in the previous row. The row number n counts the rows starting from 0 at the top. For example, row 0: 1; row 1: 1 1; row 2: 1 2 1; row 3: 1 3 3 1; and so on.

帕斯卡三角形从顶行一个1开始构建。后续每行以1开头和结尾，每个内部项由上一行中它上方的两个数相加得到。行号 n 从顶行的0开始计数。例如，第0行：1；第1行：1 1；第2行：1 2 1；第3行：1 3 3 1；以此类推。

The numbers in Pascal’s triangle are called binomial coefficients. They can be denoted as ⁿCᵣ or C(n, r), where n is the row number and r is the position from the left starting at r = 0. For instance, in row 4 the entries are 1, 4, 6, 4, 1, which correspond to ⁴C₀, ⁴C₁, ⁴C₂, ⁴C₃, ⁴C₄.

帕斯卡三角形中的数字称为二项式系数。它们可以表示为 ⁿCᵣ 或 C(n, r)，其中 n 是行号，r 是从左边开始、从0计数的位置。例如，第4行的条目为 1, 4, 6, 4, 1，分别对应 ⁴C₀、⁴C₁、⁴C₂、⁴C₃、⁴C₄。

2. Constructing the Triangle | 构造三角形

To construct Pascal’s triangle, write down a 1 at the top. Then, for each new row, place 1 at the beginning and 1 at the end. The intermediate numbers are found by adding the two numbers above: the number above and to the left plus the number above and to the right. For row 5, add the pairs from row 4 (1 4 6 4 1): 1+4=5, 4+6=10, 6+4=10, 4+1=5, giving row 5 as 1 5 10 10 5 1.

要构造帕斯卡三角形，先在顶部写一个1。然后，对于每一新行，在开头和结尾放置1。中间的数字通过将上方两数相加得到：左上方数字加上右上方数字。对于第5行，将第4行 (1 4 6 4 1) 的相邻数对相加：1+4=5，4+6=10，6+4=10，4+1=5，得到第5行为 1 5 10 10 5 1。

The triangle is symmetric because each row reads the same forwards and backwards. This symmetry reflects the combinatorial identity ⁿCᵣ = ⁿCₙ₋ᵣ.

三角形是对称的，因为每一行正读和反读相同。这种对称性反映了组合恒等式 ⁿCᵣ = ⁿCₙ₋ᵣ。

3. Binomial Coefficients and nCr Notation | 二项式系数与 nCr 符号

In A‑level mathematics, the binomial coefficient ⁿCᵣPublished by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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