Pascal's Triangle Calculator

Understanding Pascal's Triangle

What is Pascal's Triangle?

Pascal's Triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. Each number is the sum of the two numbers directly above it.

Formula

The elements of Pascal's Triangle are typically denoted \(C(n,k)\), where \(n\) is the row number and \(k\) is the position in the row (both starting from 0).

Formula: \[C(n,k) = \frac{n!}{k!(n-k)!}\]

Where:

• \(n!\) is the factorial of \(n\)
• \(k!\) is the factorial of \(k\)
• \((n-k)!\) is the factorial of \((n-k)\)

Calculation Steps

Let's calculate the 4th row of Pascal's Triangle:

1. For \(C(3,0)\): \(\frac{3!}{0!(3-0)!} = \frac{6}{1 \cdot 6} = 1\)
2. For \(C(3,1)\): \(\frac{3!}{1!(3-1)!} = \frac{6}{1 \cdot 2} = 3\)
3. For \(C(3,2)\): \(\frac{3!}{2!(3-2)!} = \frac{6}{2 \cdot 1} = 3\)
4. For \(C(3,3)\): \(\frac{3!}{3!(3-3)!} = \frac{6}{6 \cdot 1} = 1\)

Example and Visual Representation

This diagram illustrates the first 5 rows of Pascal's Triangle. Each number is the sum of the two numbers directly above it.