Pascal's Triangle
Pascal's Triangle is a triangular array where each number is the sum of the two numbers directly above it. It encodes binomial coefficients, combinatorics, and the coefficients of binomial expansions. Named after Blaise Pascal (1623–1662) though known much earlier.
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Tip: The triangle was known in China (Yang Hui, 13th century), Persia (Omar Khayyam, 11th century), and India (Pingala, 2nd century BCE) — centuries before Pascal. He's credited because of his comprehensive treatise on its properties.
- 1Row 0: 1 | Row 1: 1, 1 | Row 2: 1, 2, 1 | Row 3: 1, 3, 3, 1
- 2Entry C(n,k) = entry in row n, position k = n! / (k!(n−k)!)
- 3Binomial expansion: (a+b)^n coefficients are row n of the triangle
- 4Sum of row n = 2^n; diagonal sums give Fibonacci numbers
(x+y)^4=1x⁴ + 4x³y + 6x²y² + 4xy³ + 1y⁴Coefficients: Row 4 = 1,4,6,4,1
Combinations C(5,2)=10Row 5, position 2 of Pascal's Triangle
| Row | Values | Sum |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1, 1 | 2 |
| 2 | 1, 2, 1 | 4 |
| 3 | 1, 3, 3, 1 | 8 |
| 4 | 1, 4, 6, 4, 1 | 16 |
| 5 | 1, 5, 10, 10, 5, 1 | 32 |
| 6 | 1, 6, 15, 20, 15, 6, 1 | 64 |
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Fun Fact
Pascal's Triangle contains an infinite number of mathematical patterns including triangular numbers, Fibonacci numbers, powers of 11, and the Sierpinski triangle fractal (visible when odd/even entries are colored differently).
References
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