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How is Pascal’s triangle constructed?
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Blaise Pascal
Blaise Pascal, oil painting by an anonymous artist, c. 1650; in the Musée Carnavalet–Histoire de Paris, Paris. Pascal laid the foundation for probability theory.
Although Pascal’s triangle is named after the French philosopher and scientist Blaise Pascal, it was known to earlier mathematicians in China and the Islamic world.12
The triangle is constructed using a simple additive process:
- Edges: Place a 1 along both the left and right edges of the triangle.1
- Interior: Fill each interior position by calculating the sum of the two numbers located directly above it to the left and right.12
Starting with a single 1 at the top, this method generates successive rows:
* 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 10 10 5 1
This arrangement provides the binomial coefficients used to expand binomial expressions such as .12