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生成杨辉三角(帕斯卡三角)的指定行。

Row (n)

Position (k)

Binom

Row Sum64

Central20

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公式

Pascal's Triangle

Binomial Coefficient

C(n, k) = n! / (k! × (n-k)!)

This is the entry in row n, position k of Pascal's triangle (both starting from 0).

Properties

Each entry equals the sum of the two entries above it

Row sums: each row sums to 2^n

Symmetry: C(n, k) = C(n, n-k)

The entries give the coefficients of (a+b)^n

计算示例

Find the entry at row 6, position 2 of Pascal's triangle.

01C(6, 2) = 6! / (2! × 4!)
02= 720 / (2 × 24)
03= 720 / 48
04= 15
05Row 6: 1, 6, 15, 20, 15, 6, 1

常见问题

What is Pascal's triangle?

Pascal's triangle is a triangular array where each entry is the sum of the two entries above it. Row n contains the binomial coefficients C(n,0) through C(n,n).

How is Pascal's triangle related to the binomial theorem?

The entries in row n give the coefficients when expanding (a+b)^n. For example, row 3 is 1,3,3,1, and (a+b)³ = a³ + 3a²b + 3ab² + b³.

What patterns exist in Pascal's triangle?

Column 0 is all 1s, column 1 gives natural numbers, column 2 gives triangular numbers. The triangle also contains Fibonacci numbers along certain diagonals.

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